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Diffstat (limited to 'Linear_Algebra_by_Jim_Hefferon')
55 files changed, 765 insertions, 0 deletions
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.11/Ex1_11.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.11/Ex1_11.R new file mode 100644 index 00000000..a005839b --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.11/Ex1_11.R @@ -0,0 +1,18 @@ +#Example 1.11,page 6
+#package used: matlib
+#installation run command : install.packages("matlib")
+#package repo : https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,0,0,-1,-2,1,0,0,1,0,2,0,2,1,1),ncol=4)
+b <- c(0,4,0,5)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-1)
+Ab <- rowswap(Ab,2,3)
+Ab <- rowadd(Ab,3,4,-2)
+Ab
+#Back-substitution gives w = 1, z = 2 , y = -1, and x = -1.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.12/Ex1_12.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.12/Ex1_12.R new file mode 100644 index 00000000..e7c4bf79 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.12/Ex1_12.R @@ -0,0 +1,17 @@ +#Example 1.12 Section I. Solving Linear Systems page7
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,2,3,1,2),ncol = 2)
+b <- c(1,-3,-2)
+#for this problem we cannot use normal method because the number of equations is more than number of variables
+#so we use gaussian elimination technique.
+gaussianElimination(A, b, tol = sqrt(.Machine$double.eps), verbose = FALSE,
+ latex = FALSE, fractions = FALSE)
+# result shows that one of the equations is redundant, here x=-2,y=1
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.13/Ex1_13.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.13/Ex1_13.R new file mode 100644 index 00000000..a7d7f3c3 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.13/Ex1_13.R @@ -0,0 +1,18 @@ +#Example 1.13,page 7
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,2,3,1,2),ncol = 2)
+b <- c(1,-3,0)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-2)
+Ab <- rowadd(Ab,1,3,-2)
+Ab <- rowadd(Ab,2,3,-(4/5))
+Ab
+#the echelon form shows that the system is inconsistent,hence the solution set is empty
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.3/Ex1_3.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.3/Ex1_3.R new file mode 100644 index 00000000..663ef17b --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.3/Ex1_3.R @@ -0,0 +1,6 @@ +#Example 1.3
+#Section I. Solving Linear Systems,page 3
+A <- matrix(c(3,-1,2,1),ncol = 2)
+b <- c(7,6)
+x <- solve(A) %*% b
+x
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.4/Ex1_4.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.4/Ex1_4.R new file mode 100644 index 00000000..7a1b7ebc --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.4/Ex1_4.R @@ -0,0 +1,22 @@ +#Example 1.4,Section I. Solving Linear Systems
+#page 3
+#package used: matlib
+#installation run command : install.packages("matlib")
+#package repo : https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(0,1,1/3,0,5,2,3,-2,0),ncol = 3,nrow = 3)
+b <- c(9,2,3)
+Ab <- cbind(A,b)
+Ab <- rowswap(Ab,1,3)
+Ab <- rowmult(Ab,1,3)
+Ab <- rowmult(Ab,1,-1)
+Ab <- rowadd(Ab,1,2,1)
+Ab
+#from Ab (X3=3)(since drom row3,3*X3=9)
+#from Ab row2 -1*X2-2*x3=-7,so X2=1
+#from Ab row1 X1+6*x2=9,so X1 = 3
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.7/Ex1_7.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.7/Ex1_7.R new file mode 100644 index 00000000..86158a6f --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.7/Ex1_7.R @@ -0,0 +1,20 @@ +#Example 1.7,page nO:5
+#package used: matlib
+#installation run command : install.packages("matlib")
+#package repo : https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,1,1,-1,-2,0,3,-1),ncol = 3)
+b <- c(0,3,3)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-2)
+Ab <- rowadd(Ab,1,3,-1)
+Ab <- rowadd(Ab,2,3,-1)
+Ab
+#from row3 : -4z=0,so z=0
+#from row2 : -3y+3z=3,so y=-1
+#from row1 : x+y=0,so x=1
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.8/Ex1_8.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.8/Ex1_8.R new file mode 100644 index 00000000..c4f0cdcf --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.8/Ex1_8.R @@ -0,0 +1,17 @@ +#Example 1.8,page 5
+#package used: matlib
+#installation run command : install.packages("matlib")
+#package repo : https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(40,-50,15,25),ncol = 2)
+b <- c(100,50)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,1.25)
+Ab
+#from row2 : 43.75c = 175,so c= 4
+#from row1 : 40h+15c=100,so h=1
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.9/Ex1_9.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.9/Ex1_9.R new file mode 100644 index 00000000..25e09a21 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX1.9/Ex1_9.R @@ -0,0 +1,21 @@ +#Example 1.9,chapter1 linear systems page 6
+#Example showing gauss method to reduce given system of linear equations
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,3,1,4,6,1,-3,-5),ncol = 3)
+b <-c(9,1,0)
+Ab <- cbind(A,b)
+Ab <-rowadd(Ab,1,2,-2)
+Ab <-rowadd(Ab,1,3,-3)
+Ab <-rowadd(Ab,2,3,-(3/2))
+Ab
+#from Ab row3: z=3
+#from Ab row2: y=-1
+#from Ab row1: x=7
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.1/Ex1_2_1.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.1/Ex1_2_1.R new file mode 100644 index 00000000..0e4eac30 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.1/Ex1_2_1.R @@ -0,0 +1,20 @@ +#Example 2.1,chapter1,section 1.2,page 13
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(2,1,3,0,-1,-1,1,-1,0),ncol=3)
+b <- c(3,1,4)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-0.5)
+Ab <- rowadd(Ab,1,3,-1.5)
+Ab <- rowadd(Ab,2,3,-1)
+Ab
+#from the result we can see that the system doesnot have a unique solution.
+#We can represent the solution set by representing the variables that lead(x,y)by the variable that does not lead(z).
+#so solution set {((3/2)-(1/2)z,(1/2)-(3/2)z,z)|z belongs to R}
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.12/Ex1_2_12.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.12/Ex1_2_12.R new file mode 100644 index 00000000..5c8f03e6 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.12/Ex1_2_12.R @@ -0,0 +1,7 @@ +#Example 2.12,section 1.2,chapter 1,page 18.
+#vector addition
+a <- c(2,3,1)
+b <- c(3,-1,4)
+a+b
+c <- c(1,4,-1,-3)
+7*c
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.3/Ex1_2_3.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.3/Ex1_2_3.R new file mode 100644 index 00000000..0e15eae5 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.3/Ex1_2_3.R @@ -0,0 +1,20 @@ +#Example 2.3,chapter1,section 1.2,page 13
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,0,3,0,1,1,0,-1,1,-1,6,1,-1,1,-6,-1),ncol = 4)
+b <- c(1,-1,6,1)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,3,-3)
+Ab <- rowadd(Ab,2,3,3)
+Ab <- rowadd(Ab,2,4,1)
+Ab
+#from the reduced augmented matrix Ab,we can see that only x and y are leading,while z and w are free.
+#We represent the solution set by expressing the leading variables in terms of the free variables
+#The solution set: {(2 - 2z + 2w,-1 + z - w,z,w)|z,w belongs to R}
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.5/Ex1_2_5.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.5/Ex1_2_5.R new file mode 100644 index 00000000..c37e92e8 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.5/Ex1_2_5.R @@ -0,0 +1,19 @@ +#Example 2.5,section 1.2,page 15
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,3,2,0,2,0,1,1,0,0,-1),ncol = 4)
+b <- c(1,2,4)
+Ab <- cbind(A,b) #augmented matrix
+Ab <- rowadd(Ab,1,2,-2)
+Ab <- rowadd(Ab,1,3,-3)
+Ab <- rowadd(Ab,2,3,-1)
+Ab
+#The leading variables are x, y, and w. The variable z is free.
+#although there are infinitely many solutions, the value of w doesn't vary but is constant w = -1.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.7/Ex1_2_7.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.7/Ex1_2_7.R new file mode 100644 index 00000000..cd004d7c --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX2.7/Ex1_2_7.R @@ -0,0 +1,22 @@ +#Example 2.7,chapter one,section 1.2,page 16
+#package used: matlib
+#installation run command : install.packages("matlib")
+#package repo : https://github.com/friendly/matlib
+
+#installing and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,0,1,2,1,0,0,-1,2),ncol = 3)
+b <- c(4,0,4)
+#creating augmented matrix
+Ab <- cbind(A,b)
+Ab
+#applying reduction techniques
+Ab <- rowadd(Ab,1,3,-1)
+Ab <- rowadd(Ab,2,3,2)
+Ab
+#second row: y-z=0
+#first row: x+2*y=4
+#so solution set is : {(4-2*z = 0)}
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.3/Ex1_3_3.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.3/Ex1_3_3.R new file mode 100644 index 00000000..a8612292 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.3/Ex1_3_3.R @@ -0,0 +1,24 @@ +#Example 3.3,section 1.3,chapter 1,page 24
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(3,2,4,-1),ncol = 2)
+b <- c(3,1)
+c <- c(0,0)
+Ab <- cbind(A,b) # linear system
+Ac <- cbind(A,c) #system of homogeneous equations
+#reduction of original system
+Ab <- rowadd(Ab,1,2,-2/3)
+#reduction of homogeneous system
+Ac <- rowadd(Ac,1,2,-2/3)
+#comparing both
+Ab
+Ac
+#Obviously the two reductions go in the same way.
+#We can study how to reduce a linear systems by instead studying how to reduce the associated homogeneous system.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.5/Ex1_3_5.R b/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.5/Ex1_3_5.R new file mode 100644 index 00000000..c0a2bc07 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH1/EX3.5/Ex1_3_5.R @@ -0,0 +1,18 @@ +#Example 3.5,chapter 1,section 1.3,page 25
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(7,8,0,0,0,1,1,3,-7,-5,-3,-6,0,-2,0,-1),ncol=4)
+b <- c(0,0,0,0)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-8/7)
+Ab <- rowadd(Ab,2,3,-1)
+Ab <- rowadd(Ab,2,4,-3)
+Ab <- rowadd(Ab,3,4,-5/2)
+Ab
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.16/Ex2_1_16.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.16/Ex2_1_16.R new file mode 100644 index 00000000..87990e40 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.16/Ex2_1_16.R @@ -0,0 +1,16 @@ +#Example 1.16,Section III. Basis and Dimension,page 118
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+v <- c(3,2)
+A <- matrix(c(1,1,0,2),ncol = 2)
+Av <- cbind(A,v)
+Av <- echelon(A,v,reduced = TRUE)
+Av
+#from Av,c1 = 3 and c2 = -1=2.
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.6/Ex2_1_6.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.6/Ex2_1_6.R new file mode 100644 index 00000000..938d80aa --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.6/Ex2_1_6.R @@ -0,0 +1,19 @@ +#Chapter 2.
+#Section II. Linear Independence
+#Example 1.6,page 103
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+#c1(40 15) + c2(-50 25) = (0 0),check if {(40 15),(-50 25)} is linearly independent
+A <- matrix(c(40,15,-50,25),ncol = 2)
+b <- c(0,0)
+Ab <- cbind(A,b)
+Ab <- rowadd(Ab,1,2,-15/40)
+Ab
+#from Ab,Both c1 and c2 are zero. So the only linear relationship between the two given row vectors is the trivial relationship.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.9/Ex2_1_9.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.9/Ex2_1_9.R new file mode 100644 index 00000000..77f1a7b4 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX1.9/Ex2_1_9.R @@ -0,0 +1,8 @@ +#Chapter 2.
+#Section II. Linear Independence
+#Example 1.9,page 104
+v1 <- c(3,4,5)
+v2 <- c(2,9,2)
+v3 <- c(4,18,4)
+(0*v1)+(2*v2)-1*v3
+#the set S = {v1,v2,v3} is linearly dependent because this is a relationship where not all of the scalars are zero
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.10/Ex2_3_10.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.10/Ex2_3_10.R new file mode 100644 index 00000000..80b9f9b1 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.10/Ex2_3_10.R @@ -0,0 +1,13 @@ +#Example 3.10,Section III. Basis and Dimension,page 131
+#package used pracma
+#install package using command: install.packages("pracma")
+
+#installation and loading library
+#install.packages("pracma")
+library("pracma")
+
+#program
+A <- matrix(c(1,2,1,3,6,3,1,3,1,6,16,6),ncol = 4,nrow = 3)
+#finding row reduced echelon form("using gauss-jordan reduction")
+rref(A)
+#Thus, for a reduced echelon form matrix we can find bases for the row and column spaces in essentially the same way, by taking the parts of the matrix, the rows or columns, containing the leading entries.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.2/Ex2_3_2.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.2/Ex2_3_2.R new file mode 100644 index 00000000..be64a7ba --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.2/Ex2_3_2.R @@ -0,0 +1,18 @@ +#Example 3.2,Section III. Basis and Dimension,page 127
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(2,4,3,6),ncol = 2)
+#Rowspace(A) is this subspace of the space of two-component row vectors
+#{c1.(2 3) + c2.(4 6) | c1,c2 belongs to R}
+#simplifying A
+A <- rowadd(A,1,2,-2)
+A
+#From the simplified matrix,the second row vector is linearly dependent on the first and so we can simplify the above description to
+#{c.(2 3)|c belongs to R}
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.5/Ex2_3_5.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.5/Ex2_3_5.R new file mode 100644 index 00000000..1344c076 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.5/Ex2_3_5.R @@ -0,0 +1,20 @@ +#Example 3.5,Section III. Basis and Dimension,page 128
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,1,2,3,4,0,1,1,5),ncol = 3)
+#From any matrix, we can produce a basis for the row space by
+#performing Gauss's Method and taking the nonzero rows of the resulting echelon form matrix
+#simplifying to echelon form
+A <- rowadd(A,1,2,-1)
+A <- rowadd(A,1,3,-2)
+A <- rowadd(A,2,3,6)
+A
+#on simplification:produces the basis h(1 3 1); (0 1 0); (0 0 3)i for the row space. This is a basis
+#for the row space of both the starting and ending matrices, since the two row spaces are equal.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.7/Ex2_3_7.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.7/Ex2_3_7.R new file mode 100644 index 00000000..8044a89c --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.7/Ex2_3_7.R @@ -0,0 +1,20 @@ +#Example 3.7,Section III. Basis and Dimension,page 129
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,2,0,4,3,3,1,0,7,8,2,4),ncol = 3)
+#to get a basis for the column space, temporarily turn the columns into rows and reduce.
+A <- t(A)
+A <- rowadd(A,1,2,-3)
+A <- rowadd(A,1,3,-7)
+A <- rowadd(A,2,3,-2)
+#Now turn the rows back to columns
+A <- t(A)
+A
+#The result is a basis for the column space of the given matrix.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.9/Ex2_3_9.R b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.9/Ex2_3_9.R new file mode 100644 index 00000000..5f009c37 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH2/EX3.9/Ex2_3_9.R @@ -0,0 +1,18 @@ +#Example 3.5,Section III. Basis and Dimension,page 130
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+#To get a basis for the span of {x^2 + x^4,2x^2 + 3x^4,-x^2 - 3x^4 } in the space row4
+A <- matrix(c(0,0,0,0,0,0,1,2,-1,0,0,0,1,3,-3),ncol = 5,nrow = 3)
+#applying gauss method
+A <- rowadd(A,1,2,-2)
+A <- rowadd(A,1,3,1)
+A <- rowadd(A,2,3,2)
+A
+#we get the basis (x^2 + x^4,x^4)
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.1/Ex3_1_1.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.1/Ex3_1_1.R new file mode 100644 index 00000000..fc57c156 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.1/Ex3_1_1.R @@ -0,0 +1,13 @@ +#Example 1.1,Section I. page 166
+a <- c(1,2)
+a <-t(a) #a becomes row vector
+b <-t(a) #b becomes column vector
+c <-c(3,4)
+c <-t(c)
+d <-t(c)
+a+c
+b+d
+#these corresponding vectors add to corresponding totals
+5*a
+5*b
+#correspondence respecting scalar multiplication
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.11/Ex3_1_11.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.11/Ex3_1_11.R new file mode 100644 index 00000000..7293511e --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.11/Ex3_1_11.R @@ -0,0 +1,7 @@ +#Example 1.11,Section II. page 186
+#given map: h(1,0)=(-1,1) and h(0,1)=(-4,4)
+#h(3,-2)=h(3*(1,0)-2*(0,1))=3*h(1,0)-2*h(0,1)
+a <- c(-1,1) # h(1,0)
+b <- c(-4,4) # h(0,1)
+3*a-2*b
+#the value of h on this argument is a direct consequence of the value of h on the basis vectors.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.9/Ex3_1_9.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.9/Ex3_1_9.R new file mode 100644 index 00000000..85fb511c --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX1.9/Ex3_1_9.R @@ -0,0 +1,8 @@ +#Example 1.9,Section I. page 170
+#space P5 of polynomials of degree 5 or less and the map f that sends a polynomial p(x) to p(x - 1).
+#under this map x^2 ->(x-1)^2 = x^2-2x+1 and x^3+2x -> (x-1)^3+2(x-1) = x^3-3x^2+5x-3.
+curve(x^2,from = -1000,to=1000)
+curve((x-1)^2,from = -1000,to=1000)
+curve(x^3,from = -1000,to=1000)
+curve((x-1)^3,from = -1000,to=1000)
+#from these plots we can say that this map is an automorphism of this space.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.10/Ex3_2_10.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.10/Ex3_2_10.R new file mode 100644 index 00000000..7919e0e4 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.10/Ex3_2_10.R @@ -0,0 +1,13 @@ +#Example2.10,section III:Computing linear maps,chapter3,page 219
+#package used pracma
+#install package using command: install.packages("pracma")
+
+#installation and loading library
+#install.packages("pracma")
+library("pracma")
+
+#program
+A <- matrix(c(1,0,2,3),ncol = 2)
+Rank(A)
+#Any map from R^2 to P1 represented with respect to any pair of bases by the above matrix;
+#is nonsingular because this matrix has rank two.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.11/Ex3_2_11.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.11/Ex3_2_11.R new file mode 100644 index 00000000..513a692e --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.11/Ex3_2_11.R @@ -0,0 +1,4 @@ +#Example2.11,section III:Computing linear maps,chapter3,page 220
+A <- matrix(c(1,3,2,6),ncol=2)
+det(A)
+#map g: V -> W represented by above matrix is singular because this matrix is singular.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.5/Ex3_2_5.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.5/Ex3_2_5.R new file mode 100644 index 00000000..dd7fe300 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX2.5/Ex3_2_5.R @@ -0,0 +1,14 @@ +#Example2.5,section III:Computing linear maps,chapter3,page 218
+#package used pracma
+#install package using command: install.packages("pracma")
+
+#installation and loading library
+#install.packages("pracma")
+library("pracma")
+
+#program
+A <- matrix(c(1,1,0,0,2,2,0,0,2,1,3,2),ncol = 3)
+Rank(A)
+#Any map represented by above matrix must have three-dimensional domain and a four-dimensional codomain.
+#Since the rank of this matrix is found to be 2 by above code;
+#Any map represented by this matrix has a two-dimensional range space.
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX3.1.11/Ex3_3_1_11.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX3.1.11/Ex3_3_1_11.R new file mode 100644 index 00000000..2f98aaf6 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX3.1.11/Ex3_3_1_11.R @@ -0,0 +1,11 @@ +#Example1.11,section III:Computing linear maps,chapter3,page 211
+a <- matrix(c(1,2,0,0,-1,3),ncol=3)
+b <- c(2,-1,1)
+a %*% b
+#the above result can also be obtained by the method
+x <- matrix(c(1,2),ncol = 1)
+y <- matrix(c(0,0),ncol = 1)
+z <- matrix(c(-1,3),ncol = 1)
+#splitting the matrix a into component columns and now multiplying
+2*x-1*y+1*z
+#we see that both methods are equal.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.1/Ex3_4_1_1.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.1/Ex3_4_1_1.R new file mode 100644 index 00000000..c59f039a --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.1/Ex3_4_1_1.R @@ -0,0 +1,7 @@ +#Example1.1,section IV:Matrix Operations,chapter3,page 224
+#Let f : V -> W be a linear function represented with respect to some bases by this matrix.
+f <- matrix(c(1,1,0,1),ncol = 2)
+#find the map that is the scalar multiple 5f: V -> W.
+5*f
+#Changing from the map f to the map 5f has the effect on the representation of the output vector of multiplying each entry by 5.
+#Therefore, going from the matrix representing f to the one representing 5f means multiplying all the matrix entries by 5.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.2/Ex3_4_1_2.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.2/Ex3_4_1_2.R new file mode 100644 index 00000000..2d290db6 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.1.2/Ex3_4_1_2.R @@ -0,0 +1,7 @@ +#Example1.2,section IV:Matrix Operations,chapter3,page 225
+f <- matrix(c(1,2,3,0),ncol = 2)
+g <- matrix(c(-2,2,-1,4),ncol = 2)
+#two linear maps with the same domain and codomain f; g: R^2 -> R^2 are represented with respect to bases B and D by these matrices.
+#find effect of f + g on the map
+f+g
+#if f does v -> u and g does v -> w then f + g is the function whose action is v -> u + w.
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.4/Ex3_4_2_4.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.4/Ex3_4_2_4.R new file mode 100644 index 00000000..f0ffa844 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.4/Ex3_4_2_4.R @@ -0,0 +1,4 @@ +#Example2.4,section IV:Matrix Operations,chapter3,page 229
+A <- matrix(c(2,4,8,0,6,2),ncol = 2)
+B <- matrix(c(1,5,3,7),ncol = 2)
+A %*% B
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.5/Ex3_4_2_5.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.5/Ex3_4_2_5.R new file mode 100644 index 00000000..50f9767e --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.5/Ex3_4_2_5.R @@ -0,0 +1,4 @@ +#Example2.5,section IV:Matrix Operations,chapter3,page 229
+A <- matrix(c(1,3,2,4),ncol = 2)
+B <- matrix(c(-1,2,0,-2),ncol=2)
+A %*% B
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.6/Ex3_4_2_6.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.6/Ex3_4_2_6.R new file mode 100644 index 00000000..643cb0f1 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.2.6/Ex3_4_2_6.R @@ -0,0 +1,4 @@ +#Example2.6,section IV:Matrix Operations,chapter3,page 230
+A <- matrix(c(1,0,1,1,1,0),ncol=2)
+B <- matrix(c(4,5,6,7,8,9,2,3),ncol = 4)
+A %*% B
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.3.15/Ex3_4_3_15.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.3.15/Ex3_4_3_15.R new file mode 100644 index 00000000..c615c719 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.3.15/Ex3_4_3_15.R @@ -0,0 +1,8 @@ +#Example3.15,section IV.3:Mechanics of Matrix Multiplication,chapter3,page 241
+#A permutation matrix is square and is all 0's except for a single 1 in each row and column.
+A <- matrix(c(0,1,0,0,0,1,1,0,0),ncol = 3) #permutation matrix
+B <- matrix(c(1,4,7,2,5,8,3,6,9),ncol = 3)
+#From the left these matrices permute rows.
+A %*% B
+#From the right they permute columns.
+B %*% A
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.4.9/Ex3_4_4_9.R b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.4.9/Ex3_4_4_9.R new file mode 100644 index 00000000..17781554 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH3/EX4.4.9/Ex3_4_4_9.R @@ -0,0 +1,15 @@ +#Example4.9,section IV.4:Inverses,chapter3,page 251
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+# Augmented matrix
+A <- matrix(c(0,1,1,3,0,-1,-1,1,0),ncol = 3)
+B <- matrix(c(1,0,0,0,1,0,0,0,1),ncol = 3)
+AB <- cbind(A,B)
+echelon(A,B)
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.10/Ex4_1_10.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.10/Ex4_1_10.R new file mode 100644 index 00000000..41570377 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.10/Ex4_1_10.R @@ -0,0 +1,15 @@ +#Example 1.10,chapter 4,Section III. Laplace's Formula,page 356
+#package used matlib v0.9.1 and pracma
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+t <- matrix(c(1,2,1,0,1,0,4,-1,1),ncol = 3)
+#finding inverse: if T has an inverse, if |T| != 0, then T^-1 = (1/|T|) adj(T)
+a <- adjoint(t)
+b <- 1/det(t)
+i <- b*a
+i
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.4/Ex4_1_4.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.4/Ex4_1_4.R new file mode 100644 index 00000000..c42d8991 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.4/Ex4_1_4.R @@ -0,0 +1,14 @@ +#Example 1.4,chapter 4,Section III. Laplace's Formula,page 354
+#package used matlib v0.9.1 and pracma
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+T <- matrix(c(1,4,7,2,5,8,3,6,9),ncol = 3)
+T12 <- cofactor(T,1,2)
+T22 <- cofactor(T,2,2)
+T12
+T22
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.6/Ex4_1_6.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.6/Ex4_1_6.R new file mode 100644 index 00000000..156a2053 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.6/Ex4_1_6.R @@ -0,0 +1,17 @@ +#Example 1.6,chapter 4,Section III. Laplace's Formula,page 355\
+#package used matlib v0.9.1 and pracma
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+t <- matrix(c(1,4,7,2,5,8,3,6,9),ncol = 3)
+x <- det(t)
+#computing the determinant by expanding along the first row,
+y <- t[1,1]*cofactor(t,1,1)+t[1,2]*cofactor(t,1,2)+t[1,3]*cofactor(t,1,3)
+#computing the determinant by expanding along the second row,
+z <- t[2,1]*cofactor(t,2,1)+t[2,2]*cofactor(t,2,2)+t[2,3]*cofactor(t,2,3)
+all.equal(x,y)
+all.equal(x,z)
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.7/Ex4_1_7.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.7/Ex4_1_7.R new file mode 100644 index 00000000..e770d76f --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX1.7/Ex4_1_7.R @@ -0,0 +1,13 @@ +#Example 1.7,chapter 4,Section III. Laplace's Formula,page 355
+#package used matlib v0.9.1 and pracma
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+t <- matrix(c(1,2,3,5,1,-1,0,1,0),ncol = 3)
+#computing the determinant by expanding along the third column.
+y <- t[1,3]*cofactor(t,1,3)+t[2,3]*cofactor(t,2,3)+t[3,3]*cofactor(t,3,3)
+y
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.5/Ex4_2_5.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.5/Ex4_2_5.R new file mode 100644 index 00000000..2d4cdeab --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.5/Ex4_2_5.R @@ -0,0 +1,20 @@ +#Example 2.5,chapter4,Section I. Definition,page 325
+#package used matlib v0.9.1
+#install package using command: install.packages("matlib")
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(2,4,0,2,4,-3,6,3,5),ncol=3)
+b <- det(A)
+#determinant by normal ,ethod
+A <- rowadd(A,1,2,-2)
+A <- rowswap(A,2,3)
+A
+#reducing with gaussian reduction now multiplying the diagonal terms,keeping in mind the sign change due to row swap to find determinant.
+c <- -1*A[1,1]*A[2,2]*A[3,3]
+all.equal(b,c)
+#so the determinant by both techniques are the same
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.6/Ex4_2_6.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.6/Ex4_2_6.R new file mode 100644 index 00000000..a00f9279 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX2.6/Ex4_2_6.R @@ -0,0 +1,16 @@ +#Example 2.6,chapter4,Section I. Definition,page 326
+#package used matlib v0.9.1 and pracma
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+
+#program
+A <- matrix(c(1,0,0,0,0,1,0,1,1,1,0,0,3,4,5,1),ncol = 4)
+A <- rowadd(A,2,4,-1)
+A <- rowswap(A,3,4)
+A
+#multiplying diagonal terms and multiplying it with -1 to compensate for rowswap
+-1*A[1,1]*A[2,2]*A[3,3]*A[4,4]
+
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.1/Ex4_3_1.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.1/Ex4_3_1.R new file mode 100644 index 00000000..3e45a8cf --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.1/Ex4_3_1.R @@ -0,0 +1,9 @@ +#Example3.1,chapter4,section1,page 329
+X <- matrix(c(4,-2,2,6),ncol = 2)
+#scalars come out of each row separately,not from the entire matrix at once.
+#so X will become
+A <- matrix(c(2,-1,1,3),ncol = 2)
+#with scalars 2 & 2 coming out of rows 1 and 2 respectively
+c <- det(X)
+d <- 4*det(A)
+all.equal(c,d)
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.4/Ex4_3_4.R b/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.4/Ex4_3_4.R new file mode 100644 index 00000000..29df6c40 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH4/EX3.4/Ex4_3_4.R @@ -0,0 +1,11 @@ +#Example3.4,chapter 4,page 330
+A <- matrix(c(2,4,1,3),ncol = 2)
+#using multinlinearity property to break up the matrix
+a <- matrix(c(2,4,0,0),ncol = 2)
+b <- matrix(c(2,0,0,3),ncol = 2)
+c <- matrix(c(0,4,1,0),ncol = 2)
+d <- matrix(c(0,0,1,3),ncol = 2)
+#verifying the property
+x <- det(A)
+y <- det(a)+det(b)+det(c)+det(d)
+all.equal(x,y)
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX1.3/Ex5_1_3.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX1.3/Ex5_1_3.R new file mode 100644 index 00000000..3d1d36dc --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX1.3/Ex5_1_3.R @@ -0,0 +1,12 @@ +#Example 1.3,chapter 5,Section II. Similarity,page 390
+#package used matlib v0.9.1
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+P <- matrix(c(2,1,1,1),ncol = 2)
+T <- matrix(c(2,1,-3,-1),ncol = 2)
+#finding similar matrix of T
+T1 <- P %*% T %*% Inverse(P)
+T1
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.10/Ex5_2_10.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.10/Ex5_2_10.R new file mode 100644 index 00000000..63487d6c --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.10/Ex5_2_10.R @@ -0,0 +1,16 @@ +#Example 2.10,chapter 5,scetion III.2,page 414
+#package used matlib v0.9.1
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+N <- matrix(c(0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0),ncol=4)
+P <- matrix(c(1,0,1,0,0,2,1,0,1,1,1,0,0,0,0,1),ncol = 4)
+A <- P %*% N %*% Inverse(P)
+A
+#The new matrix,A is nilpotent; its fourth power is the zero matrix.
+x <- P %*% N^4 %*% Inverse(P)
+#since (PNP^-1)^4 = P * N^4 *P^-1
+y <- det(x)
+all.equal(y,0)
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.17/Ex5_2_17.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.17/Ex5_2_17.R new file mode 100644 index 00000000..e6436638 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.17/Ex5_2_17.R @@ -0,0 +1,21 @@ +#Example 2.17,chapter 5,scetion III.2,page 419
+#package used matlib v0.9.1
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+M <- matrix(c(1,1,-1,-1),ncol = 2)
+#finding nilpotent index
+A <- matrix(c(0,0,0,0),ncol = 2)
+count <- 1
+Y <- M
+repeat{
+ Y <- Y %*% M
+ if (all.equal(Y,A)){
+ print(count+1)
+ break()
+ }
+ count= count+1
+}
+
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.18/Ex5_2_18.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.18/Ex5_2_18.R new file mode 100644 index 00000000..960e5899 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.18/Ex5_2_18.R @@ -0,0 +1,20 @@ +#Example 2.18,chapter 5,scetion III.2,page 420
+#package used matlib v0.9.1
+#Github reposiory of matlib :https://github.com/friendly/matlib
+
+#installation and loading library
+#install.packages("matlib")
+library("matlib")
+M <- matrix(c(0,1,-1,0,1,0,0,1,1,0,0,0,1,0,-1,0,0,-1,0,1,0,0,1,0,-1),ncol = 5)
+#finding nilpotent index
+A <- matrix(c(0),ncol = 5,nrow = 5)
+count <- 1
+Y <- M
+repeat{
+ Y <- Y %*% M
+ if (all.equal(Y,A)== TRUE){
+ print(count+1)
+ break()
+ }
+ count=count+1
+}
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2.2/Ex5_2_2_2.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2.2/Ex5_2_2_2.R new file mode 100644 index 00000000..b78f52e9 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2.2/Ex5_2_2_2.R @@ -0,0 +1,7 @@ +#Example 2.2,chapter 5,section II.2 Diagonalizability,page 393
+#matrix: T is diagonalizable if there is a nonsingular P such that PTP-1 is diagonal.
+T <- matrix(c(4,1,-2,1),ncol = 2)
+P <- matrix(c(-1,1,2,-1),ncol = 2)
+#checking whether diagonizable
+P %*% T %*% Inverse(P) # diagonal matrix
+# so matrix P is diagonizable
\ No newline at end of file diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2/Ex5_2_2.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2/Ex5_2_2.R new file mode 100644 index 00000000..cf70b449 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.2/Ex5_2_2.R @@ -0,0 +1,12 @@ +#Example 2.2,chapter 5,Section I. Complex Vector Spaces,page 387
+a <- complex(real = 1,imaginary = 1)
+b <- complex(real = 0,imaginary = 1)
+c <- complex(real = 2,imaginary = -0)
+d <- complex(real = -2,imaginary = 3)
+e <- complex(real = 1,imaginary = 0)
+f <- complex(real = 0,imaginary = 3)
+g <- complex(real = 1,imaginary = -0)
+h <- complex(real = 0,imaginary = -1)
+A <- matrix(c(a,b,c,d),ncol = 2)
+B <- matrix(c(e,f,g,h),ncol = 2)
+A %*% B
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.4/Ex5_2_4.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.4/Ex5_2_4.R new file mode 100644 index 00000000..ed6d2f8a --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX2.4/Ex5_2_4.R @@ -0,0 +1,15 @@ +#Example 2.4,chapter 5,section IV.2,page 432
+#package used pracma v1.9.9
+#installing and loading library
+#install.packages("pracma")
+library("pracma")
+T <- matrix(c(2,1,-1,4),ncol=2)
+a <- eigen(T)
+a$values
+#so T has only the single eigenvalue 3.
+I <- matrix(c(1,0,0,1),ncol = 2)
+T-(3*I)
+# so for this,the only eigenvalue is 0 and T -3I is nilpotent.
+#to ease this computation we find nulspaces
+x <- nullspace(T)
+x
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.19/Ex5_3_19.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.19/Ex5_3_19.R new file mode 100644 index 00000000..6b619a24 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.19/Ex5_3_19.R @@ -0,0 +1,4 @@ +#Example3.19,chapter 5,section II.3 Eigenvalues and Eigenvectors,page 404
+T <- matrix(c(2,0,-4,-2,1,8,2,1,3),ncol = 3)
+a <- eigen(T)
+a$values
diff --git a/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.6/Ex5_3_6.R b/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.6/Ex5_3_6.R new file mode 100644 index 00000000..9bf70023 --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/CH5/EX3.6/Ex5_3_6.R @@ -0,0 +1,5 @@ +#Example3.6,chapter 5,section II.3 Eigenvalues and Eigenvectors,page 398
+T <- matrix(c(2,0,0,0),ncol = 2)
+a <-eigen(T)
+a$values
+a$vectors
diff --git a/Linear_Algebra_by_Jim_Hefferon/README.txt b/Linear_Algebra_by_Jim_Hefferon/README.txt new file mode 100644 index 00000000..5bd5c46e --- /dev/null +++ b/Linear_Algebra_by_Jim_Hefferon/README.txt @@ -0,0 +1,18 @@ +About The Contributor + +Contributed By: Ajay Ragh +Course: Bachelor Of Technology +Branch: Information Technology +College/Institute/Organization: Apj Abdul Kalam Technological University + +About The Book + +Book: Linear Algebra +Author: Jim Hefferon +Publisher: Orthogonal Publishing L3c +Year Of Publication: 2017 +ISBN: 978-1944325039 +Edition: 3 + + +Textbook Companion Project By FOSSEE, IIT Bombay |