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-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/README.md8
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/README.md17
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gifbin664757 -> 1828325 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gifbin29814 -> 283795 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gifbin117597 -> 1370273 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/README.md14
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gifbin47411 -> 433985 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gifbin2770965 -> 5401504 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/README.md23
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gifbin2042905 -> 2051994 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gifbin551768 -> 1238872 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gifbin2263059 -> 5251558 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gifbin2145303 -> 4953394 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gifbin5123074 -> 11846180 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gifbin722341 -> 1583937 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gifbin5660857 -> 13732426 bytes
-rw-r--r--FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gifbin95828 -> 952963 bytes
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py79
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py174
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py33
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py191
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file1_orthogonal.py26
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_OrthonormalBasis.py82
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_sum_of_projections_part1.py133
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file3_sum_of_projections_part2.py173
-rwxr-xr-xFSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py77
-rw-r--r--FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py75
27 files changed, 1013 insertions, 92 deletions
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/README.md
new file mode 100644
index 0000000..a62369d
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/README.md
@@ -0,0 +1,8 @@
+**file1_directional_deriv**
+![file1_directional_deriv](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file1_directional_deriv.gif)
+
+**file2_gradient**
+![file2_gradient](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file2_gradient.gif)
+
+**file3_gradient_level_curves**
+![file3_gradient_level_curves](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/directional-derivatives/gifs/file3_gradient_level_curves.gif)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/README.md
new file mode 100644
index 0000000..0e6e8d3
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/README.md
@@ -0,0 +1,17 @@
+**file1_multivar_func_examples**
+![file1_multivar_func_examples](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file1_multivar_func_examples.gif)
+
+**file2_multivariable_func_respresentation**
+![file2_multivariable_func_respresentation](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif)
+
+**file3_sphere**
+![file3_sphere](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file3_sphere.gif)
+
+**file4_vectorvf_sine**
+![file4_vectorvf_sine](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif)
+
+**file5_vectorvf_helix**
+![file5_vectorvf_helix](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file5_vectorvf_helix.gif)
+
+**file6_derivative_vectorvf**
+![file6_derivative_vectorvf](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif
index 40add0f..8c4506c 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file2_multivariable_func_respresentation.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif
index 4f6b931..215459e 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file4_vectorvf_sine.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif
index a94de90..9ea94e4 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-functions/gifs/file6_derivative_vectorvf.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/README.md
new file mode 100644
index 0000000..c01ddc5
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/README.md
@@ -0,0 +1,14 @@
+**file1_epsilon_delta_defn**
+![file1_epsilon_delta_defn](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file1_epsilon_delta_defn.gif)
+
+**file2_limit_approach_point**
+![file2_limit_approach_point](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif)
+
+**file3_limit_approach_point_3d**
+![file3_limit_approach_point_3d](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif)
+
+**file4_limit_different_point**
+![file4_limit_different_point](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file4_limit_different_point.gif)
+
+**file5_continuity_func**
+![file5_continuity_func](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file5_continuity_func.gif)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif
index 830b6f1..3abd596 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file2_limit_approach_point.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif
index 4bccf8c..3e87cdd 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/multivariable-limits-and-continuity/gifs/file3_limit_approach_point_3d.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/README.md b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/README.md
new file mode 100644
index 0000000..c62dd51
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/README.md
@@ -0,0 +1,23 @@
+**file1_partial_deriv_gas_law**
+![file1_partial_deriv_gas_law](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif)
+
+**file2_partial_deriv_hill**
+![file2_partial_deriv_hill](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif)
+
+**file3_partial_deriv_defn**
+![file3_partial_deriv_defn](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif)
+
+**file4_partial_deriv_example**
+![file4_partial_deriv_example](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif)
+
+**file5_partial_deriv_func_2maximas**
+![file5_partial_deriv_func_2maximas](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif)
+
+**file6_clariant_rule**
+![file6_clariant_rule](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file6_clariant_rule.gif)
+
+**file7_partial_deriv_clariant_rule**
+![file7_partial_deriv_clariant_rule](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif)
+
+**file8_chain_rule**
+![file8_chain_rule](https://github.com/nishanpoojary/FSF-mathematics-python-code-archive/blob/master/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file8_chain_rule.gif)
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif
index 560a7c0..8fdb80f 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file1_partial_deriv_gas_law.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif
index f4c3f49..3c758ff 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file2_partial_deriv_hill.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif
index e0e42db..c66b3fa 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file3_partial_deriv_defn.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif
index 30682cb..d2bf541 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file4_partial_deriv_example.gif
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diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif
index aa74437..db7f4f8 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file5_partial_deriv_func_2maximas.gif
Binary files differ
diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif
index ecef499..32d5e92 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/partial-derivatives/gifs/file7_partial_deriv_clariant_rule.gif
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diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif
index af9e536..9576b4a 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file3_parabola_example.gif
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diff --git a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif
index 9d24688..a22f1b8 100644
--- a/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif
+++ b/FSF-2020/calculus-of-several-variables/multivariable-functions-and-paritial-derivatives/scalar-functions/gifs/file7_neural_nets.gif
Binary files differ
diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py
new file mode 100755
index 0000000..dd4b8d4
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file1_projections.py
@@ -0,0 +1,79 @@
+from manimlib.imports import *
+
+class Projections(GraphScene):
+ CONFIG = {
+ "x_min": -6,
+ "x_max": 6,
+ "y_min": -4,
+ "y_max": 4,
+ "graph_origin" : ORIGIN ,
+ }
+ def construct(self):
+
+ self.setup_axes(animate=True)
+
+ XTD = self.x_axis_width/(self.x_max-self.x_min)
+ YTD = self.y_axis_height/(self.y_max-self.y_min)
+
+ arrow_a = Arrow(start = ORIGIN, end = 4*XTD*RIGHT)
+ arrow_a.scale(1.2)
+ arrow_a.set_color(DARK_BLUE)
+ arrow_b = Arrow(start = ORIGIN, end = 2*YTD*UP+2*XTD*RIGHT)
+ arrow_b.scale(1.3)
+ arrow_b.set_color(DARK_BLUE)
+ self.play(ShowCreation(arrow_a), ShowCreation(arrow_b))
+
+ text = TextMobject(r"Consider 2 linearly independent vectors $a$ and $b$")
+ text.set_color(DARK_BLUE)
+ text.scale(0.6)
+ text.move_to(3*YTD*UP+5*XTD*LEFT)
+ text_a = TextMobject("a")
+ text_a.move_to(0.4*YTD*DOWN+3*XTD*RIGHT)
+ text_a.set_color(DARK_BLUE)
+ text_b = TextMobject("b")
+ text_b.move_to(1.5*YTD*UP+RIGHT*XTD)
+ text_b.set_color(DARK_BLUE)
+
+ self.play(Write(text),Write(text_a), Write(text_b))
+ self.wait()
+
+ arrow_b_copy = Arrow(start = ORIGIN, end = 2*YTD*UP+2*XTD*RIGHT)
+ arrow_b_copy.scale(1.25)
+
+ arrow_p = Arrow(start = ORIGIN, end = 2*XTD*RIGHT)
+ arrow_p.scale(1.5)
+ arrow_p.set_color(GOLD_E)
+
+ text_p = TextMobject("p")
+ text_p.move_to(0.25*DOWN+RIGHT)
+ text_p.set_color(GOLD_E)
+
+ self.play(FadeOut(text), Transform(arrow_b_copy,arrow_p), FadeOut(text_a), FadeOut(text_b))
+ text = TextMobject(r"$p$ is the projection of $b$ on $a$")
+ text.set_color(GOLD_E)
+ text.move_to(3*UP+4*LEFT)
+ text.scale(0.8)
+ self.play(Write(text),Write(text_p))
+ self.wait()
+
+ self.play(FadeIn(text_a), FadeIn(text_b))
+
+ arrow_o = Arrow(start = 2*XTD*RIGHT, end = 2*YTD*UP+2*XTD*RIGHT)
+ arrow_o.scale(1.5)
+ arrow_o.set_color(GREEN_E)
+
+ text_o = TextMobject("b-p")
+ text_o.move_to(UP*YTD+2.7*XTD*RIGHT)
+ text_o.set_color(GREEN_E)
+
+ self.play(ShowCreation(arrow_o))
+ self.play(FadeOut(text),Write(text_o))
+
+ text = TextMobject(r"Observe, ($b-p$) is orthogonal to $a$")
+ text.set_color(GREEN_E)
+ text.move_to(2*DOWN+4*LEFT)
+ text.scale(0.8)
+ self.play(Write(text))
+ self.wait(2)
+
+ self.play(FadeOut(self.axes), FadeOut(arrow_a), FadeOut(arrow_b), FadeOut(arrow_b_copy), FadeOut(arrow_o), FadeOut(text_a), FadeOut(text_b), FadeOut(text_o), FadeOut(text_p), FadeOut(text)) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py
new file mode 100644
index 0000000..640b8be
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file2_orthonormal.py
@@ -0,0 +1,174 @@
+from manimlib.imports import *
+
+from manimlib.imports import *
+
+class Orthogonal(ThreeDScene):
+ def construct(self):
+
+ text = TextMobject(r"These are two Orthogonal Basis $\alpha_{1}$ and $\alpha_{2}$")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*(DOWN+RIGHT))
+ self.play(Write(text))
+
+ axes = ThreeDAxes()
+ self.play(ShowCreation(axes))
+ self.move_camera(phi=60*DEGREES,theta=45*DEGREES,run_time=3)
+ self.begin_ambient_camera_rotation(rate=0.3)
+
+ line1 = Line(start = ORIGIN,end = 1*RIGHT)
+ line1.set_color(DARK_BLUE)
+ tip1 = Polygon(RIGHT,0.8*RIGHT-0.2*DOWN,0.8*RIGHT-0.2*UP)
+ tip1.set_opacity(1)
+ tip1.set_fill(DARK_BLUE)
+ tip1.set_color(DARK_BLUE)
+
+ arrow2 = Line(start = ORIGIN,end = 1*UP)
+ arrow2.set_color(DARK_BLUE)
+ tip2 = Polygon(UP,0.8*UP-0.2*RIGHT,0.8*UP-0.2*LEFT)
+ tip2.set_opacity(1)
+ tip2.set_fill(DARK_BLUE)
+ tip2.set_color(DARK_BLUE)
+ arrow2.set_color(DARK_BLUE)
+
+ self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2))
+ self.wait()
+
+ a_line = Line(start = ORIGIN,end = 2*UP+2*RIGHT+[0,0,2])
+ a_line.set_color(GOLD_E)
+ a_tip = Polygon(2*UP+2*RIGHT+[0,0,2],2*UP+1.6*RIGHT+[0,0,1.8],1.6*UP+2*RIGHT+[0,0,1.8])
+ a_tip.set_opacity(1)
+ a_tip.set_fill(GOLD_E)
+ a_tip.set_color(GOLD_E)
+
+ a_line_c1 = Line(start = ORIGIN,end = 2*UP+2*RIGHT+[0,0,2])
+ a_line_c1.set_color(GOLD_E)
+ a_tip_c1 = Polygon(2*UP+2*RIGHT+[0,0,2],2*UP+1.6*RIGHT+[0,0,1.8],1.6*UP+2*RIGHT+[0,0,1.8])
+ a_tip_c1.set_opacity(1)
+ a_tip_c1.set_fill(GOLD_E)
+ a_tip_c1.set_color(GOLD_E)
+
+ self.play(FadeOut(text), ShowCreation(a_line), ShowCreation(a_tip), ShowCreation(a_line_c1), ShowCreation(a_tip_c1))
+
+ text = TextMobject(r"There is a vector $\beta_3$")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*(DOWN+RIGHT))
+ self.play(Write(text))
+ self.wait()
+ self.play(FadeOut(text))
+
+ p_line1 = Line(start = ORIGIN,end = 2*RIGHT)
+ p_line1.set_color(GOLD_E)
+ p_tip1 = Polygon(RIGHT,0.8*RIGHT+0.2*DOWN,0.8*RIGHT+0.2*UP)
+ p_tip1.move_to(2*RIGHT)
+ p_tip1.set_opacity(1)
+ p_tip1.set_fill(GOLD_E)
+ p_tip1.set_color(GOLD_E)
+
+ self.play(Transform(a_line_c1,p_line1),Transform(a_tip_c1,p_tip1))
+
+ text = TextMobject(r"Take projection of $\beta_3$ on $\alpha_1$")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*(DOWN+RIGHT))
+ self.play(Write(text))
+ self.wait()
+ self.play(FadeOut(text))
+
+ o_line1 = Line(start = 2*RIGHT,end = 2*UP+2*RIGHT+[0,0,2])
+ o_line1.set_color(GREEN_E)
+ o_tip1 = Polygon(2*UP+2*RIGHT+[0,0,2],1.8*UP+2*RIGHT+[0,0,1.8]+0.2*RIGHT,1.8*UP+2*RIGHT+[0,0,1.8]-0.2*RIGHT)
+ o_tip1.set_opacity(1)
+ o_tip1.set_fill(GREEN_E)
+ o_tip1.set_color(GREEN_E)
+
+ a_line1 = Line(start = ORIGIN,end = 2*UP+[0,0,2])
+ a_line1.set_color(GREEN_E)
+ a_tip1 = Polygon(2*UP+[0,0,2],1.8*UP+[0,0,1.8]+0.2*RIGHT,1.8*UP+[0,0,1.8]-0.2*RIGHT)
+ a_tip1.set_opacity(1)
+ a_tip1.set_fill(GREEN_E)
+ a_tip1.set_color(GREEN_E)
+
+ a_line1_c1 = Line(start = ORIGIN,end = 2*UP+[0,0,2])
+ a_line1_c1.set_color(GREEN_E)
+ a_tip1_c1 = Polygon(2*UP+[0,0,2],1.8*UP+[0,0,1.8]+0.2*RIGHT,1.8*UP+[0,0,1.8]-0.2*RIGHT)
+ a_tip1_c1.set_opacity(1)
+ a_tip1_c1.set_fill(GREEN_E)
+ a_tip1_c1.set_color(GREEN_E)
+
+ text = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$)")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*(DOWN+RIGHT))
+ self.play(Write(text))
+ self.play(ShowCreation(o_line1), ShowCreation(o_tip1))
+ self.wait(2)
+ self.play(FadeOut(a_line_c1), FadeOut(a_tip_c1),
+ FadeOut(a_line), FadeOut(a_tip),
+ Transform(o_line1,a_line1), Transform(o_tip1,a_tip1))
+
+ self.wait()
+ self.play(FadeOut(text))
+
+ p_arrow2 = Line(start = ORIGIN,end = 2*UP)
+ p_arrow2.set_color(GOLD_E)
+ p_tip2 = Polygon(2*UP,1.8*UP+0.2*RIGHT,1.8*UP+0.2*LEFT)
+ p_tip2.set_opacity(1)
+ p_tip2.set_fill(GOLD_E)
+ p_tip2.set_color(GOLD_E)
+ p_arrow2.set_color(GOLD_E)
+
+ last_a = Line(start = 2*UP,end = [0,2,2])
+ last_a.set_color(GOLD_E)
+ last_a_tip = Polygon([0,0,2],[0,0,1.8]+0.2*RIGHT,[0,0,1.8]+0.2*LEFT)
+ last_a_tip.move_to([0,2,2])
+ last_a_tip.set_opacity(1)
+ last_a_tip.set_fill(GOLD_E)
+ last_a_tip.set_color(GOLD_E)
+
+ self.wait()
+ text = TextMobject(r"Take projection on $\alpha_2$")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*(DOWN+RIGHT))
+ self.play(Write(text))
+ self.play(Transform(a_line1_c1,p_arrow2),Transform(a_tip1_c1,p_tip2))
+ self.wait()
+ self.play(FadeOut(text))
+
+ text = TextMobject(r"$\beta_3$-(projection of $\beta_3$ on $\alpha_1$ + projection of $\beta_3$ on $\alpha_2$)")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN+2*RIGHT)
+ self.play(Write(text))
+ self.play(ShowCreation(o_line1), ShowCreation(o_tip1))
+ self.wait(2)
+ self.play(ShowCreation(last_a_tip), ShowCreation(last_a))
+ self.wait()
+ self.play(FadeOut(text))
+
+ larrow3 = Line(start = ORIGIN,end = [0,0,2])
+ larrow3.set_color(GOLD_E)
+ ltip3 = Polygon([0,0,2],[0,0,1.8]+0.2*RIGHT,[0,0,1.8]+0.2*LEFT)
+ ltip3.set_opacity(1)
+ ltip3.set_fill(GOLD_E)
+ ltip3.set_color(GOLD_E)
+ self.wait()
+ self.play(FadeOut(o_line1), FadeOut(o_tip1), FadeOut(a_line1_c1), FadeOut(a_tip1_c1), Transform(last_a,larrow3), Transform(last_a_tip,ltip3))
+
+ text = TextMobject(r"Normalisation of the orthogonal vector")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN+2*RIGHT)
+ self.play(Write(text))
+
+ arrow3 = Line(start = ORIGIN,end = [0,0,1])
+ arrow3.set_color(DARK_BLUE)
+ tip3 = Polygon([0,0,1],[0,0,0.8]-0.2*RIGHT,[0,0,0.8]-0.2*LEFT)
+ tip3.set_opacity(1)
+ tip3.set_fill(DARK_BLUE)
+ tip3.set_color(DARK_BLUE)
+ self.play(Transform(last_a,arrow3), Transform(last_a_tip,tip3))
+
+ self.wait(3) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py
new file mode 100644
index 0000000..ccd23c9
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Gram Schmidt Orthonormalization Process/file_introduction.py
@@ -0,0 +1,33 @@
+from manimlib.imports import *
+
+class Orthonormal(Scene):
+ def construct(self):
+ Centre = DOWN
+ arrow_1 = Arrow(start = Centre+ORIGIN,end = Centre+1.414*(UP+RIGHT))
+ arrow_2 = Arrow(start = Centre+ORIGIN,end = Centre+2*UP)
+ arrow_1.scale(1.35)
+ arrow_2.scale(1.35)
+ text = TextMobject("This is a set of linearly independent vectors")
+ text.scale(0.75)
+ text.move_to(3*UP+3*LEFT)
+ text.set_color(PURPLE_E)
+ arrow_1.set_color(PURPLE_E)
+ arrow_2.set_color(PURPLE_E)
+ self.play(Write(text))
+ self.play(ShowCreation(arrow_1), ShowCreation(arrow_2))
+ self.wait(2)
+ text1 = TextMobject("After we apply Gram-Schmidt Orthogonalization Process to set of linearly independent vectors")
+ text1.scale(0.6)
+ text1.move_to(3*UP+2*LEFT)
+ text1.set_color(GREEN)
+ arrow_a = Arrow(start = Centre+ORIGIN,end = Centre+0.707*(UP+RIGHT))
+ arrow_a.set_color(GREEN)
+ arrow_a.scale(2)
+ self.play(Transform(text,text1))
+ self.wait(2)
+ self.play(Transform(arrow_1,arrow_a))
+ arrow_b = Arrow(start = Centre+ORIGIN,end = Centre+0.707*(UP+LEFT))
+ arrow_b.set_color(GREEN)
+ arrow_b.scale(2)
+ self.play(Transform(arrow_2,arrow_b))
+ self.wait(2) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py
new file mode 100644
index 0000000..85a2983
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Linear-Transformations-(Linear-Maps)/file_before_matrix.py
@@ -0,0 +1,191 @@
+from manimlib.imports import *
+
+class linear(GraphScene):
+
+ CONFIG = {
+ "x_min": -5,
+ "x_max": 5,
+ "y_min": -5,
+ "y_max": 5,
+ "graph_origin": ORIGIN+RIGHT,
+ "x_labeled_nums": list(range(-5, 6)),
+ "y_labeled_nums": list(range(-5, 6)),
+ "x_axis_width": 7,
+ "y_axis_height": 7,
+ }
+
+ def construct(self):
+
+ XTD = self.x_axis_width/(self.x_max- self.x_min)
+ YTD = self.y_axis_height/(self.y_max- self.y_min)
+
+ self.setup_axes(animate = True)
+ heading = TextMobject(r"$T(x,y) = T(x+2y,x-y)$")
+ heading.move_to(UP*3+LEFT*4)
+ self.play(Write(heading))
+ self.wait()
+
+ before = TextMobject("Before Linear Transformation")
+ before.set_color(DARK_BLUE)
+ before.move_to(3*UP+4*RIGHT)
+ before.scale(0.75)
+ dot1 = Dot().shift(self.graph_origin+1*XTD*RIGHT+1*YTD*UP)
+ dot2 = Dot().shift(self.graph_origin+2*XTD*RIGHT+1*YTD*UP)
+ dot1.set_color(DARK_BLUE)
+ dot2.set_color(DARK_BLUE)
+ p1 = TextMobject(r"$P_1$")
+ p1.set_color(DARK_BLUE)
+ p1.move_to(self.graph_origin+1*XTD*RIGHT+2*YTD*UP)
+ p2 = TextMobject(r"$P_2$")
+ p2.set_color(DARK_BLUE)
+ p2.move_to(self.graph_origin+2*XTD*RIGHT+2*YTD*UP)
+
+ after = TextMobject("After applying Linear Transformation")
+ after.set_color(RED)
+ after.move_to(3*UP+4.5*RIGHT)
+ after.scale(0.75)
+ dot3 = Dot().shift(self.graph_origin+3*XTD*RIGHT+0*YTD*UP)
+ dot4 = Dot().shift(self.graph_origin+4*XTD*RIGHT+1*YTD*UP)
+ dot3.set_color(RED)
+ dot4.set_color(RED)
+ p3 = TextMobject(r"$P_3$")
+ p3.set_color(RED)
+ p3.move_to(self.graph_origin+3*XTD*RIGHT-1.1*YTD*UP)
+ p4 = TextMobject(r"$P_4$")
+ p4.set_color(RED)
+ p4.move_to(self.graph_origin+4*XTD*RIGHT+2*YTD*UP)
+
+ tp1 = TextMobject(r"$T(P_1) = P_3$")
+ tp1.set_color(RED)
+ tp1.move_to(DOWN+4*LEFT)
+ tp2 = TextMobject(r"$T(P_2) = P_4$")
+ tp2.set_color(RED)
+ tp2.move_to(2*DOWN+4*LEFT)
+
+ self.play(Write(before), ShowCreation(dot1), ShowCreation(dot2),Write(p1), Write(p2))
+ self.wait(3)
+ self.play(Transform(before,after), Transform(dot1,dot3), Transform(dot2,dot4), Transform(p2,p4), Transform(p1,p3), Write(tp1), Write(tp2))
+ self.wait(3)
+
+
+class withgrid(LinearTransformationScene):
+ def construct(self):
+
+ heading = TextMobject(r"$T(x,y) = (x+2y,x-y)$")
+ heading.move_to(UP*2.5+LEFT*4)
+ self.play(Write(heading))
+ self.wait()
+
+ before = TextMobject("Before Linear Transformation")
+ before.set_color(DARK_BLUE)
+ before.move_to(3.5*UP+4*RIGHT)
+ before.scale(0.75)
+ dot1 = Dot().shift(1*RIGHT+1*UP)
+ dot2 = Dot().shift(2*RIGHT+1*UP)
+ dot1.set_color(DARK_BLUE)
+ dot2.set_color(DARK_BLUE)
+
+ dot1_c = Dot(radius = 0.05).shift(1*RIGHT+1*UP)
+ dot2_c = Dot(radius = 0.05).shift(2*RIGHT+1*UP)
+ dot1_c.set_color(RED)
+ dot2_c.set_color(RED)
+ self.add_transformable_mobject(dot1_c)
+ self.add_transformable_mobject(dot2_c)
+
+ p1 = TextMobject(r"$P_1$")
+ p1.set_color(DARK_BLUE)
+ p1.move_to(1*RIGHT+1.5*UP)
+ p2 = TextMobject(r"$P_2$")
+ p2.set_color(DARK_BLUE)
+ p2.move_to(2*RIGHT+1.5*UP)
+
+ after = TextMobject("After applying Linear Transformation")
+ after.set_color(RED)
+ after.move_to(3.5*UP+3.5*RIGHT)
+ after.scale(0.75)
+ dot3 = Dot().shift(3*RIGHT+0*UP)
+ dot4 = Dot().shift(4*RIGHT+1*UP)
+ dot3.set_color(RED)
+ dot4.set_color(RED)
+ p3 = TextMobject(r"$P_3$")
+ p3.set_color(RED)
+ p3.move_to(3*RIGHT-0.6*UP)
+ p4 = TextMobject(r"$P_4$")
+ p4.set_color(RED)
+ p4.move_to(4*RIGHT+1.5*UP)
+
+ self.play(Write(before), ShowCreation(dot1), ShowCreation(dot2),Write(p1), Write(p2))
+ self.wait(3)
+ matrix = [[1,2],[1,-1]]
+ self.apply_matrix(matrix)
+ self.play(Transform(before,after), Transform(p2,p4), Transform(p1,p3))
+ self.play(Transform(before,after))
+ self.wait(3)
+
+ ending = TextMobject(r"$T(\left[\begin{array}{c}x \\ y\end{array}\right]) = \left[\begin{array}{c} x+2y \\ x-y\end{array}\right]$")
+ ending.move_to(UP*2+LEFT*4)
+ self.play(Transform(heading,ending))
+ self.wait()
+
+from manimlib.imports import *
+class ThreeDExplanation(ThreeDScene):
+
+ def construct(self):
+
+ text = TextMobject(r"$T(x,y) = (x+y,x-y,x+2y)$")
+ text.scale(0.75)
+ text.move_to(UP*2.5+LEFT*4)
+ self.add_fixed_in_frame_mobjects(text)
+ self.play(Write(text))
+ self.wait()
+
+ before = TextMobject("Before Linear Transformation")
+ self.add_fixed_in_frame_mobjects(before)
+ before.set_color(GREEN_E)
+ before.move_to(3.5*UP+4*RIGHT)
+ before.scale(0.75)
+ dot1 = Dot().shift(1*RIGHT+1*UP)
+ dot2 = Dot().shift(2*RIGHT+1*UP)
+ dot3 = Dot().shift(1*RIGHT+1*UP)
+ dot1.set_color(GREEN_E)
+ dot2.set_color(GREEN_E)
+ dot3.set_color(GREEN_E)
+ self.play(ShowCreation(before))
+
+ dot1_c = Dot(radius = 0.05).shift(1*RIGHT+1*UP)
+ dot2_c = Dot(radius = 0.05).shift(0*RIGHT+2*UP)
+ dot3_c = Dot(radius = 0.05).shift(1*RIGHT-1*UP)
+ dot1_c.set_color(RED)
+ dot2_c.set_color(RED)
+ dot3_c.set_color(RED)
+
+ axes = ThreeDAxes(x_min = -7,x_max=7,y_min=-4,y_max=4,z_min=-4,z_max=4)
+ self.play(ShowCreation(axes))
+ self.move_camera(distance = 100, phi=30*DEGREES,theta=45*DEGREES,run_time=3)
+
+ self.begin_ambient_camera_rotation(rate=0.3)
+ self.wait(1)
+ self.stop_ambient_camera_rotation()
+
+ plane = NumberPlane()
+ self.play(ShowCreation(dot1),ShowCreation(dot3),ShowCreation(dot2),ShowCreation(plane))
+
+ self.play(FadeOut(before))
+ after = TextMobject("After applying Linear Transformation")
+ self.add_fixed_in_frame_mobjects(after)
+ after.set_color(RED)
+ after.move_to(3.5*UP+3.5*RIGHT)
+ after.scale(0.75)
+
+ matrix = [[1,1],[1,-1],[2,1]]
+ self.play(FadeOut(dot1),FadeOut(dot2),FadeOut(dot3),ApplyMethod(plane.apply_matrix,matrix),ApplyMethod(dot1_c.apply_matrix,matrix),ApplyMethod(dot3_c.apply_matrix,matrix),ApplyMethod(dot2_c.apply_matrix,matrix))
+ self.begin_ambient_camera_rotation(rate=0.3)
+ self.wait(3)
+ self.stop_ambient_camera_rotation()
+
+ ending = TextMobject(r"$T(\left[\begin{array}{c}x \\ y\end{array}\right]) = \left[\begin{array}{c} x+y \\ x-y \\ x+2y \end{array}\right]$")
+ ending.scale(0.75)
+ ending.move_to(-UP*2+LEFT*4)
+ self.play(Transform(text,ending))
+ self.add_fixed_in_frame_mobjects(ending)
+ self.wait(9)
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file1_orthogonal.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file1_orthogonal.py
index b400f93..a5d96f5 100755
--- a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file1_orthogonal.py
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file1_orthogonal.py
@@ -4,31 +4,37 @@ class Orthogonal(ThreeDScene):
def construct(self):
axes = ThreeDAxes()
self.play(ShowCreation(axes))
- self.move_camera(phi=30*DEGREES,theta=-45*DEGREES,run_time=3)
- line1 = Line(start = ORIGIN,end = -3*LEFT)
+ self.move_camera(phi=60*DEGREES,theta=45*DEGREES,run_time=3)
+
+ text = TextMobject(r"$\hat{i}$",r"$\hat{j}$",r"$\hat{k}$")
+ text[0].move_to(0.7*DOWN+0.8*LEFT)
+ text[1].move_to(0.75*DOWN+0.7*RIGHT)
+ text[2].move_to(0.75*UP+0.4*RIGHT)
+ self.add_fixed_in_frame_mobjects(text)
+ self.play(Write(text))
+
+ line1 = Line(start = ORIGIN,end = RIGHT)
line1.set_color(DARK_BLUE)
- tip1 = Polygon(-LEFT,-0.8*LEFT-0.2*DOWN,-0.8*LEFT-0.2*UP)
- tip1.move_to(-3*LEFT)
+ tip1 = Polygon(-0.95*LEFT,-0.8*LEFT-0.1*DOWN,-0.8*LEFT-0.1*UP)
tip1.set_opacity(1)
tip1.set_fill(DARK_BLUE)
tip1.set_color(DARK_BLUE)
- arrow2 = Line(start = ORIGIN,end = -3*UP)
+ arrow2 = Line(start = ORIGIN,end = UP)
arrow2.set_color(DARK_BLUE)
- tip2 = Polygon(DOWN,0.8*DOWN-0.2*RIGHT,0.8*DOWN-0.2*LEFT)
- tip2.move_to(3*DOWN)
+ tip2 = Polygon(0.95*UP,0.8*UP-0.1*RIGHT,0.8*UP-0.1*LEFT)
tip2.set_opacity(1)
tip2.set_fill(DARK_BLUE)
tip2.set_color(DARK_BLUE)
arrow2.set_color(DARK_BLUE)
- arrow3 = Line(start = ORIGIN,end = [0,0,3])
+ arrow3 = Line(start = ORIGIN,end = [0,0,1])
arrow3.set_color(DARK_BLUE)
- tip3 = Polygon([0,0,3],[0,0,2.8]-0.2*RIGHT,[0,0,2.8]-0.2*LEFT)
+ tip3 = Polygon([0,0,0.95],[0,0,0.8]-0.1*RIGHT,[0,0,0.8]-0.1*LEFT)
tip3.set_opacity(1)
tip3.set_fill(DARK_BLUE)
tip3.set_color(DARK_BLUE)
self.play(ShowCreation(line1), ShowCreation(tip1), ShowCreation(arrow2), ShowCreation(tip2), ShowCreation(arrow3), ShowCreation(tip3))
- self.wait() \ No newline at end of file
+ self.wait()
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_OrthonormalBasis.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_OrthonormalBasis.py
deleted file mode 100644
index 0a28f22..0000000
--- a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_OrthonormalBasis.py
+++ /dev/null
@@ -1,82 +0,0 @@
-from manimlib.imports import *
-class OrthonormalBasis(GraphScene):
- CONFIG = {
- "x_min" : -6,
- "x_max" : 6,
- "y_min" : -4,
- "y_max" : 4,
- "graph_origin" : ORIGIN ,
-}
-
- def construct(self):
- self.setup_axes(animate=True)
-
- XTD = self.x_axis_width/(self.x_max-self.x_min)
- YTD = self.y_axis_height/(self.y_max-self.y_min)
-
- arrow1 = Arrow(start = ORIGIN,end = 0.707*YTD*UP+0.707*XTD*RIGHT)
- arrow1.scale(2.25)
- arrow1.set_color(DARK_BLUE)
-
- arrow2 = Arrow(start = ORIGIN,end = 0.707*YTD*UP+0.707*XTD*LEFT)
- arrow2.scale(2.25)
- arrow2.set_color(DARK_BLUE)
-
- square = Polygon(UP*0.4*YTD,0.2*(YTD*UP+XTD*RIGHT),ORIGIN,0.2*(YTD*UP+XTD*LEFT))
- square.set_color(DARK_BLUE)
- self.play(ShowCreation(arrow2), ShowCreation(arrow1), ShowCreation(square))
-
- ortho = TextMobject("Orthonormal Vectors")
- ortho.scale(0.75)
- ortho.move_to(DOWN+3*RIGHT)
- self.play(Write(ortho))
- self.wait()
- self.play(FadeOut(ortho))
-
- arrow3 = Arrow(start = ORIGIN,end = YTD*3*UP+XTD*LEFT)
- arrow3.scale(1.25)
- arrow3.set_color(GOLD_E)
- self.play(ShowCreation(arrow3))
-
- arrow4 = Arrow(start = ORIGIN,end = YTD*UP+XTD*RIGHT)
- arrow4.scale(1.8)
- arrow4.set_color(GOLD_A)
-
- arrow5 = Arrow(start = ORIGIN,end = 2*YTD*UP-2*XTD*RIGHT)
- arrow5.scale(1.3)
- arrow5.set_color(GOLD_A)
-
- self.play(ShowCreation(arrow5), ShowCreation(arrow4))
-
- self.wait()
-
- self.play(FadeOut(arrow1), FadeOut(arrow2), FadeOut(square))
-
- self.wait()
-
- text1 = TextMobject(r"$<v,v_1> v_1$")
- text1.move_to(UP+2*RIGHT)
- text1.scale(0.75)
- text2 = TextMobject(r"$<v,v_2> v_2$")
- text2.move_to(UP+3*LEFT)
- text2.scale(0.75)
-
- text3 = TextMobject("v")
- text3.move_to(YTD*3.5*UP+XTD*1.5*LEFT)
-
- self.play(Write(text1), Write(text2), Write(text3))
- self.wait()
-
- line1 = DashedLine(start = YTD*UP+XTD*RIGHT, end = YTD*3*UP+XTD*1*LEFT)
- line2 = DashedLine(start = YTD*2*UP+XTD*2*LEFT, end = YTD*3*UP+XTD*1*LEFT)
- self.play(ShowCreation(line1),ShowCreation(line2))
-
- self.wait()
-
- text = TextMobject(r"$v$ is the sum of projections","on the orthonormal vectors")
- text[0].move_to(DOWN+3.2*RIGHT)
- text[1].move_to(1.5*DOWN+3.2*RIGHT)
- self.play(Write(text))
- self.wait(2)
- self.play(FadeOut(arrow3), FadeOut(arrow4), FadeOut(arrow5), FadeOut(text1), FadeOut(text2), FadeOut(text3), FadeOut(self.axes), FadeOut(line1), FadeOut(line2))
- self.play(FadeOut(text))
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_sum_of_projections_part1.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_sum_of_projections_part1.py
new file mode 100755
index 0000000..81a0888
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file2_sum_of_projections_part1.py
@@ -0,0 +1,133 @@
+from manimlib.imports import *
+class LinearTrans(LinearTransformationScene):
+ CONFIG = {
+ "show_basis_vectors": True,
+ "basis_vector_stroke_width": 1,
+ "leave_ghost_vectors": False,
+ "show_coordinates": True,
+ }
+
+ def construct(self):
+
+ self.setup()
+
+ matrix = [[0.6,-0.8],[0.8,0.6]]
+ self.apply_matrix(matrix)
+
+ self.wait(2)
+ orthonormal = TextMobject(r"These are 2 orthonormal vectors($v_1$ and $v_2$)")
+ orthonormal.scale(0.7)
+ orthonormal.move_to(DOWN+LEFT*3.5)
+ orthonormal.add_background_rectangle()
+ v1 = TextMobject(r"$v_1$")
+ v1.scale(0.75)
+ v1.set_color(X_COLOR)
+ v1.move_to(0.75*UP+RIGHT)
+ v1.add_background_rectangle()
+ v2 = TextMobject(r"$v_2$")
+ v2.scale(0.75)
+ v2.set_color(Y_COLOR)
+ v2.move_to(0.75*UP+LEFT*1.25)
+ v2.add_background_rectangle()
+ self.play(Write(orthonormal))
+ self.play(Write(v1),Write(v2))
+ self.wait()
+ self.play(FadeOut(orthonormal), FadeOut(v1), FadeOut(v2))
+
+ arrow = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ arrow.scale(1.2)
+ arrow.set_color(BLUE)
+ arrow.apply_matrix(matrix)
+ text3 = TextMobject("v")
+ text3.move_to(3.2*UP+1.2*RIGHT)
+ text3.add_background_rectangle()
+ self.play(ShowCreation(arrow),Write(text3))
+ self.wait()
+ v_cor = TextMobject("(1,3)")
+ v_cor.move_to(3.2*UP+1.3*RIGHT)
+ v_cor.set_color(BLUE)
+ v_cor.scale(0.75)
+ v_cor.add_background_rectangle()
+ self.play(Transform(text3,v_cor))
+
+ line1 = DashedLine(start = 1*UP+3*RIGHT, end = 3*RIGHT)
+ line2 = DashedLine(start = 1*UP+3*RIGHT, end = UP)
+ line1.apply_matrix(matrix)
+ line2.apply_matrix(matrix)
+ self.play(ShowCreation(line1),ShowCreation(line2),run_time = 2)
+
+ v1 = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ v1.scale(1.2)
+ v1.set_color(BLUE)
+ v1.apply_matrix(matrix)
+ arrow1 = Arrow(start = ORIGIN,end = 3*RIGHT)
+ arrow1.scale(1.2)
+ arrow1.set_color("#6B8E23")
+ arrow1.apply_matrix(matrix)
+ self.play(Transform(v1,arrow1))
+ v1_cor = TextMobject(r"$<v,v_1> v_1$")
+ v1_cor.move_to(2.5*UP+3*RIGHT)
+ v1_cor.scale(0.75)
+ v1_cor.add_background_rectangle()
+ self.play(Write(v1_cor))
+ self.wait(0.5)
+ text1 = TextMobject(r"(1.8,2.4)")
+ text1.move_to(2.1*UP+2.5*RIGHT)
+ text1.scale(0.75)
+ text1.set_color("#6B8E23")
+ text1.add_background_rectangle()
+ self.play(Transform(v1_cor,text1))
+
+ v2 = Arrow(start = ORIGIN,end = 3*RIGHT+UP)
+ v2.scale(1.2)
+ v2.set_color("#8b0000")
+ v2.apply_matrix(matrix)
+ arrow2 = Arrow(start = ORIGIN,end = UP)
+ arrow2.scale(2.1)
+ arrow2.set_color("#8b0000")
+ arrow2.apply_matrix(matrix)
+ self.wait(0.5)
+ self.play(Transform(v2,arrow2))
+ self.wait(0.5)
+ v2_cor = TextMobject(r"$<v,v_2> v_2$")
+ v2_cor.move_to(0.75*UP+2.5*LEFT)
+ v2_cor.scale(0.75)
+ v2_cor.add_background_rectangle()
+ self.play(Write(v2_cor))
+ self.wait(0.5)
+ text2 = TextMobject(r"(-0.8,0.6)")
+ text2.move_to(0.75*UP+1.75*LEFT)
+ text2.scale(0.75)
+ text2.set_color("#8b0000")
+ text2.add_background_rectangle()
+ self.play(Transform(v2_cor,text2))
+
+ self.wait()
+
+ self.play(ApplyMethod(v2.move_to,1.4*RIGHT+2.7*UP),FadeOut(v1_cor),FadeOut(v2_cor),FadeOut(v_cor))
+
+ self.wait()
+
+ ending = TextMobject(r"$v = <v,v_1> v_1 + <v,v_2> v_2$")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending))
+
+ ending = TextMobject(r"$\left[ \begin{array} {c} 1\\ 3 \end{array}\right] = \left[ \begin{array} {c}1.8 \\ 2.4 \end{array}\right] + \left[ \begin{array} {c} -0.8\\ 0.6 \end{array}\right]$")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending))
+
+ ending = TextMobject(r"$v$ is the sum of projections on the orthonormal vectors")
+ ending.scale(0.7)
+ ending.move_to(DOWN)
+ ending.add_background_rectangle()
+ self.play(Write(ending))
+ self.wait()
+ self.play(FadeOut(ending)) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file3_sum_of_projections_part2.py b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file3_sum_of_projections_part2.py
new file mode 100644
index 0000000..9d25192
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/Orthonormal Basis/file3_sum_of_projections_part2.py
@@ -0,0 +1,173 @@
+from manimlib.imports import *
+class ThreeDExplanation(ThreeDScene):
+
+ def construct(self):
+
+ text = TextMobject("Let us consider the example discussed above again. These are the things we know:-")
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*UP)
+ self.play(Write(text))
+ self.wait(2)
+ basis = TextMobject(r"Set of Orthonormal Basis - $\left(\begin{array}{c}\frac{1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{array}\right),\left(\begin{array}{c}\frac{-1}{\sqrt{2}}\\\frac{1}{\sqrt{2}}\\0\end{array}\right),\left(\begin{array}{c}0\\0\\1\end{array}\right)$")
+ basis.scale(0.75)
+ basis.move_to(UP*1.5)
+ self.play(Write(basis))
+ v = TextMobject(r"$v_1$",r"$v_2$",r"$v_3$")
+ v[0].move_to(UP*0.5+RIGHT*0.75)
+ v[1].move_to(UP*0.5+RIGHT*2.5)
+ v[2].move_to(UP*0.5+RIGHT*4)
+ eq = TextMobject(r"$v = \left(\begin{array}{c}3\\4\\5\end{array}\right)$")
+ eq1 = TextMobject(r"$<v,v_1> = \frac{3}{\sqrt{2}} + \frac{4}{\sqrt{2}} + 0 = \frac{7}{\sqrt{2}}$")
+ eq2 = TextMobject(r"$<v,v_2> = \frac{-3}{\sqrt{2}} + \frac{4}{\sqrt{2}} + 0 =\frac{1}{\sqrt{2}}$")
+ eq3 = TextMobject(r"$<v,v_3> =  0 + 0 + 5 =5$")
+ eq.move_to(4*LEFT+DOWN)
+ eq1.move_to(0.5*DOWN+2*RIGHT)
+ eq2.move_to(1.5*DOWN+2*RIGHT)
+ eq3.move_to(2.5*DOWN+2*RIGHT)
+ self.play(Write(v))
+ self.play(Write(eq))
+ self.play(Write(eq1))
+ self.play(Write(eq2))
+ self.play(Write(eq3))
+ self.wait()
+ self.play(FadeOut(text), FadeOut(basis), FadeOut(eq), FadeOut(v), FadeOut(eq1), FadeOut(eq2), FadeOut(eq3))
+ self.wait()
+
+ text = TextMobject("These are the 3 mutually orthonormal basis of the set(", r"$v_1$, ", r"$v_2$, ", r"$v_3$",")")
+ text[1].set_color(DARK_BLUE)
+ text[2].set_color(RED)
+ text[3].set_color(YELLOW)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text))
+ self.wait()
+
+ axes = ThreeDAxes(x_min = -9,x_max=9,y_min=-9,y_max=9,z_min=-9,z_max=9)
+ self.play(ShowCreation(axes))
+ self.move_camera(distance = 100, phi=30*DEGREES,theta=45*DEGREES,run_time=3)
+ self.begin_ambient_camera_rotation(rate=0.3)
+
+ dashedline1 = DashedLine(start = -12*(UP+RIGHT), end = 12*(UP+RIGHT))
+ dashedline2 = DashedLine(start = -12*(UP+LEFT), end = 12*(UP+LEFT))
+ dashedline3 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = 3.5*UP+3.5*RIGHT)
+ dashedline4 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = 0.5*UP+0.5*LEFT)
+ dashedline5 = DashedLine(start = 4*UP+3*RIGHT+[0,0,5], end = [0,0,5])
+
+ self.play(ShowCreation(dashedline1), ShowCreation(dashedline2))
+
+ line1 = Line(start = ORIGIN,end = 0.707*RIGHT + 0.707*UP)
+ line1.set_color(DARK_BLUE)
+ tip1 = Polygon(0.707*RIGHT + 0.707*UP, 0.707*RIGHT + 0.607*UP, 0.607*RIGHT + 0.707*UP)
+ tip1.set_opacity(1)
+ tip1.set_fill(DARK_BLUE)
+ tip1.set_color(DARK_BLUE)
+ self.play(ShowCreation(line1), ShowCreation(tip1))
+
+ line2 = Line(start = ORIGIN,end = 0.707*LEFT + 0.707*UP)
+ line2.set_color(RED)
+ tip2 = Polygon(0.707*LEFT + 0.707*UP, 0.707*LEFT + 0.607*UP, 0.607*LEFT + 0.707*UP)
+ tip2.set_opacity(1)
+ tip2.set_fill(RED)
+ tip2.set_color(RED)
+
+ self.play(ShowCreation(line2), ShowCreation(tip2))
+
+ line3 = Line(start = ORIGIN,end = [0,0,1])
+ line3.set_color(YELLOW)
+ tip3 = Polygon([0,0,1],[0,0,0.8]-0.2*RIGHT,[0,0,0.8]-0.2*LEFT)
+ tip3.set_opacity(1)
+ tip3.set_fill(YELLOW)
+ tip3.set_color(YELLOW)
+ self.play(ShowCreation(line3), ShowCreation(tip3))
+ self.wait()
+
+ self.play(FadeOut(text))
+
+ text = TextMobject("Take the projection of ", r"$v$", " on the mutually orthonormal vectors")
+ text[1].set_color(GOLD_E)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text))
+ self.wait(2)
+
+ a_line = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ a_line.set_color(GOLD_E)
+ a_tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ a_tip.set_opacity(1)
+ a_tip.set_fill(GOLD_E)
+ a_tip.set_color(GOLD_E)
+
+ self.play(ShowCreation(a_line), ShowCreation(a_tip))
+ self.wait(9)
+ self.play(ShowCreation(dashedline3),ShowCreation(dashedline4),ShowCreation(dashedline5))
+ self.wait(6)
+
+ pv1 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv1.set_color(GOLD_E)
+ pv1tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv1tip.set_opacity(1)
+ pv1tip.set_fill(GOLD_E)
+ pv1tip.set_color(GOLD_E)
+
+ v1_p = Line(start = ORIGIN,end = 3.5*RIGHT + 3.5*UP)
+ v1_p.set_color(BLUE_E)
+ v1_p_tip = Polygon(3.5*RIGHT + 3.5*UP, 3.5*RIGHT + 3.4*UP, 3.4*RIGHT + 3.5*UP)
+ v1_p_tip.set_opacity(1)
+ v1_p_tip.set_fill(BLUE_E)
+ v1_p_tip.set_color(BLUE_E)
+
+ pv2 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv2.set_color(GOLD_E)
+ pv2tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv2tip.set_opacity(1)
+ pv2tip.set_fill(GOLD_E)
+ pv2tip.set_color(GOLD_E)
+
+ v2_p = Line(start = ORIGIN,end = 0.5*LEFT + 0.5*UP)
+ v2_p.set_color(RED_E)
+ v2_p_tip = Polygon(0.5*LEFT + 0.5*UP, 0.5*LEFT + 0.4*UP, 0.4*LEFT + 0.5*UP)
+ v2_p_tip.set_opacity(1)
+ v2_p_tip.set_fill(RED_E)
+ v2_p_tip.set_color(RED_E)
+
+ pv3 = Line(start = ORIGIN,end = 4*UP+3*RIGHT+[0,0,5])
+ pv3.set_color(GOLD_E)
+ pv3tip = Polygon(4*UP+3*RIGHT+[0,0,5],3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*UP+0.1*LEFT,3.6*UP+2.7*RIGHT+[0,0,4.5]+0.1*DOWN+0.1*RIGHT)
+ pv3tip.set_opacity(1)
+ pv3tip.set_fill(GOLD_E)
+ pv3tip.set_color(GOLD_E)
+
+ v3_p = Line(start = ORIGIN,end = [0,0,5])
+ v3_p.set_color(YELLOW_E)
+ v3_p_tip = Polygon([0,0,5],[0,0,4.8]+0.2*RIGHT,[0,0,4.8]+0.2*LEFT)
+ v3_p_tip.set_opacity(1)
+ v3_p_tip.set_fill(YELLOW_E)
+ v3_p_tip.set_color(YELLOW_E)
+
+ self.stop_ambient_camera_rotation()
+ self.play(Transform(pv1,v1_p), Transform(pv1tip,v1_p_tip), Transform(pv2,v2_p), Transform(pv2tip,v2_p_tip), Transform(pv3,v3_p), Transform(pv3tip,v3_p_tip))
+ self.play(FadeOut(dashedline1),
+ FadeOut(dashedline2),
+ FadeOut(dashedline3),
+ FadeOut(dashedline4),
+ FadeOut(dashedline5),
+ FadeOut(line1),
+ FadeOut(tip1),
+ FadeOut(line2),
+ FadeOut(tip2),
+ FadeOut(line3),
+ FadeOut(tip3),
+ FadeOut(text))
+
+ text = TextMobject(r"$v$ is the sum of projections on the orthonormal vectors")
+ text.set_color(GOLD_E)
+ text.scale(0.75)
+ self.add_fixed_in_frame_mobjects(text)
+ text.move_to(3*DOWN)
+ self.play(Write(text), ApplyMethod(pv2.move_to,(3.5*RIGHT + 3.5*UP+3*RIGHT+4*UP)/2), ApplyMethod(pv2tip.move_to,(3.1*RIGHT + 3.9*UP)))
+ self.play(ApplyMethod(pv3.move_to,3*RIGHT + 4*UP + [0,0,2.5]), ApplyMethod(pv3tip.move_to,(3*RIGHT + 4*UP + [0,0,4.8])))
+
+ self.wait(3) \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py
new file mode 100755
index 0000000..95d1021
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/Axb.py
@@ -0,0 +1,77 @@
+from manimlib.imports import *
+
+class Axb(Scene):
+
+ def construct(self):
+
+ text0 = TextMobject("Linear System Of Equations")
+ text1 = TextMobject(r"$x_{1}+x_{2}+x_{3} =b_{1}$")
+ text2 = TextMobject(r"$x_{1}+2x_{2}+x_{3} =b_{2}$")
+ text3 = TextMobject(r"$x_{1}+x_{2}+3x_{3} =b_{3}$")
+ text0.move_to(UP*2+LEFT*2)
+ text0.set_color(DARK_BLUE)
+ text1.move_to(UP)
+ text2.move_to(ORIGIN)
+ text3.move_to(DOWN)
+
+ text0.scale(0.75)
+ text1.scale(0.75)
+ text2.scale(0.75)
+ text3.scale(0.75)
+ self.play(Write(text0))
+ self.play(Write(text1))
+ self.play(Write(text2))
+ self.play(Write(text3))
+ self.play(ApplyMethod(text0.move_to,3*UP+LEFT*2), ApplyMethod(text1.move_to,2.5*UP), ApplyMethod(text2.move_to,2*UP), ApplyMethod(text3.move_to,1.5*UP))
+
+ A = TextMobject(r"$\left( \begin{array}{c c c} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 3 \end{array}\right) \left[ \begin{array} {c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] =$", r"$\left[ \begin{array}{c} x_{1}+x_{2}+x_{3} \\ x_{1}+2x_{2}+x_{3} \\ x_{1}+x_{2}+3x_{3} \end{array}\right]$")
+ A.scale(0.75)
+ self.play(FadeIn(A))
+
+ textA = TextMobject("A")
+ textx = TextMobject("x")
+ textb = TextMobject("Ax")
+
+ textA.move_to(DOWN+3*LEFT)
+ textx.move_to(1.1*DOWN+0.5*LEFT)
+ textb.move_to(DOWN-2*LEFT)
+
+ self.play(Write(textA), Write(textx), Write(textb))
+
+ circle1 = Circle(radius = 0.24)
+ circle2 = Circle(radius = 0.24)
+ square = Square(side_length = 0.6)
+
+ circle1.move_to(UP*0.5+LEFT*3.05)
+ circle2.move_to(UP*0.4+LEFT*0.5)
+ square.move_to(UP*0.4+RIGHT*1.3)
+
+ self.play(FadeIn(circle1), FadeIn(circle2),FadeIn(square))
+
+ self.play(ApplyMethod(circle1.move_to,UP*0.5+LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,UP*0.4+RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,UP*0.5+LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,UP*0.4+RIGHT*3.1))
+
+ self.play(ApplyMethod(circle1.move_to,LEFT*3.05), ApplyMethod(circle2.move_to,UP*0.4+LEFT*0.5), ApplyMethod(square.move_to,RIGHT*1.3))
+ self.play(ApplyMethod(circle1.move_to,LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,RIGHT*3.1))
+
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*3.05), ApplyMethod(circle2.move_to,UP*0.4+LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*1.3))
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*2.45), ApplyMethod(circle2.move_to,LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*2.2))
+ self.play(ApplyMethod(circle1.move_to,0.4*DOWN+LEFT*1.85), ApplyMethod(circle2.move_to,DOWN*0.5+LEFT*0.5), ApplyMethod(square.move_to,0.4*DOWN+RIGHT*3.1))
+
+ self.play(FadeOut(circle1), FadeOut(circle2), FadeOut(square))
+ self.play(FadeOut(A[0]), ApplyMethod(A[1].move_to,2*LEFT),ApplyMethod(textb.move_to,DOWN+1.7*LEFT), FadeOut(textx), FadeOut(textA))
+ b = TextMobject(r"$=\left[ \begin{array}{c} b_{1} \\ b_{2} \\ b_{3} \end{array}\right]$")
+ b.move_to(RIGHT)
+ textB = TextMobject("b")
+ textB.move_to(1.2*DOWN+1.1*RIGHT)
+ self.play(FadeIn(b),FadeIn(textB))
+
+ self.wait()
+
+ self.play(FadeOut(text0), FadeOut(text1), FadeOut(text2), FadeOut(text3))
+
+ axb = TextMobject("Ax = b")
+ self.play(FadeIn(axb), FadeOut(textb), FadeOut(textB), FadeOut(b), FadeOut(A[1]))
+
+ self.wait() \ No newline at end of file
diff --git a/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py
new file mode 100644
index 0000000..fb31881
--- /dev/null
+++ b/FSF-2020/linear-algebra/linear-transformations/The-Four-Fundamental-Subspaces/solution.py
@@ -0,0 +1,75 @@
+from manimlib.imports import *
+class solution(LinearTransformationScene):
+ def construct(self):
+
+ self.setup()
+ self.wait()
+
+ o = TextMobject(r"This is the original $2D$ vector space(before Linear Transformation)")
+ o.move_to(DOWN)
+ o.scale(0.75)
+ o.add_background_rectangle()
+ self.play(Write(o))
+ self.wait()
+ self.play(FadeOut(o))
+
+ A = TextMobject("Let $A$ denote the matrix the of this linear transformation.")
+ A.move_to(DOWN)
+ A.scale(0.75)
+ A.add_background_rectangle()
+ self.play(Write(A))
+ matrix = [[1,-1],[1,-1]]
+ self.apply_matrix(matrix)
+ self.wait()
+ self.play(FadeOut(A))
+
+ o = TextMobject(r"This is the transformed vector space i.e. a line ($1D$)")
+ o.move_to(DOWN)
+ o.scale(0.75)
+ o.add_background_rectangle()
+ self.play(Write(o))
+ self.wait()
+ self.play(FadeOut(o))
+
+ arrow2 = Arrow(start = ORIGIN, end = 2*DOWN+2*LEFT)
+ arrow2.set_color(DARK_BLUE)
+ arrow2.scale(1.2)
+ self.play(ShowCreation(arrow2))
+ self.wait()
+
+ o1 = TextMobject("If the vector lies in the transformed vector space")
+ o2 = TextMobject("(the line) then the solution exist")
+ o1.move_to(2*DOWN+2*RIGHT)
+ o2.move_to(2.5*DOWN+2*RIGHT)
+ o1.scale(0.75)
+ o2.scale(0.75)
+ o1.add_background_rectangle()
+ o2.add_background_rectangle()
+ self.play(Write(o1))
+ self.play(Write(o2))
+ self.wait()
+ self.play(FadeOut(o1), FadeOut(o2))
+
+ self.play(FadeOut(arrow2))
+
+ arrow1 = Arrow(start = ORIGIN, end = 2*UP+RIGHT)
+ arrow1.set_color(DARK_BLUE)
+ arrow1.scale(1.3)
+ self.play(ShowCreation(arrow1))
+ self.wait()
+
+ o1 = TextMobject("If the vector does lies in the transformed")
+ o2 = TextMobject("vector space then the does not solution exist")
+ o1.move_to(2*DOWN+2*RIGHT)
+ o2.move_to(2.5*DOWN+2*RIGHT)
+ o1.scale(0.75)
+ o2.scale(0.75)
+ o1.add_background_rectangle()
+ o2.add_background_rectangle()
+ self.play(Write(o1))
+ self.play(Write(o2))
+ self.wait()
+ self.play(FadeOut(o1), FadeOut(o2))
+
+ self.play(FadeOut(arrow1))
+ \ No newline at end of file