initial commit / add all books

93 files changed, 1555 insertions, 0 deletions

diff --git a/26/CH1/EX1.1.1/1_1_1.sce b/26/CH1/EX1.1.1/1_1_1.sce
new file mode 100755
index 000000000..0d184fe90
--- /dev/null
+++ b/26/CH1/EX1.1.1/1_1_1.sce
@@ -0,0 +1,20 @@
+disp('performing Gaussian elimination')

+a=[1 5;-2 -7]

+disp('the co-efficient matrix is:')

+disp(a)

+b=[7;-5]

+c=[a b]

+disp('the augmented matrix is:')

+disp(c)

+disp('R2=R2+2*R1')

+c(2,:)=c(2,:)+2*c(1,:)

+disp(c)

+disp('R2=(1/3)*R2')

+c(2,:)=(1/3)*c(2,:)

+disp(c)

+disp('R1=R1-5*R2')

+c(1,:)=c(1,:)-5*c(2,:)

+disp(c)

+x1=c(1,3)/c(1,1)

+x2=c(2,3)/c(2,2)

+printf('the solution is:x1=%d x2=%d',x1,x2)
\ No newline at end of file
diff --git a/26/CH1/EX1.1.13/1_1_13.sce b/26/CH1/EX1.1.13/1_1_13.sce
new file mode 100755
index 000000000..90c529311
--- /dev/null
+++ b/26/CH1/EX1.1.13/1_1_13.sce
@@ -0,0 +1,22 @@
+disp('the augmented matrix is')

+a=[1 0 -3 8;2 2 9 7;0 1 5 -2]

+disp(a)

+disp('R2=R2-2*R1')

+a(2,:)=a(2,:)-2*a(1,:)

+disp(a)

+disp('interchange R2 and R3')

+a([2,3],:)=a([3,2],:)

+disp(a)

+disp('R3=R3-2*R2')

+a(3,:)=a(3,:)-2*a(2,:)

+disp(a)

+disp('R3=(1/5)*R3')

+a(3,:)=(1/5)*a(3,:)

+disp(a)

+disp('R2=R2-5*R3 and R1=R1+3*R3')

+a(2,:)=a(2,:)-5*a(3,:)

+a(1,:)=a(1,:)+3*a(3,:)

+disp(a)

+s=[a(1,4);a(2,4);a(3,4)]

+disp('solution is')

+disp(s)
\ No newline at end of file
diff --git a/26/CH1/EX1.1.19/1_1_19.sce b/26/CH1/EX1.1.19/1_1_19.sce
new file mode 100755
index 000000000..0b7c2e672
--- /dev/null
+++ b/26/CH1/EX1.1.19/1_1_19.sce
@@ -0,0 +1,8 @@
+disp('the augmented matrix for h=2')

+a=[1 2 4;3 6 8]

+disp(a)

+disp('R2-2*R1')

+a(2,:)=a(2,:)-3*a(1,:)

+disp(a)

+disp('from R3 we get 0=-4')

+disp('hence, if h=2 no solution, else solution exists')
\ No newline at end of file
diff --git a/26/CH1/EX1.1.25/1_1_25.sce b/26/CH1/EX1.1.25/1_1_25.sce
new file mode 100755
index 000000000..552d996fb
--- /dev/null
+++ b/26/CH1/EX1.1.25/1_1_25.sce
@@ -0,0 +1,14 @@
+disp('the co-efficient matrix is:')

+a=[1 -4 7;0 3 -5;-2 5 -9]

+disp(a)

+disp('let g,h,k be the constants on RHS')

+disp('R3=R3+2*R1')

+a(3,:)=a(3,:)+2*a(1,:)

+disp(a)

+disp('the constants on RHS are:g,h,k+2g')

+disp('R3=R3+R2')

+a(3,:)=a(3,:)+a(2,:)

+disp(a)

+disp('the constants on RHS are:g,h,k+2g+h')

+disp('for solution to exist')

+disp('from R3:k+2g+h=0')
\ No newline at end of file
diff --git a/26/CH1/EX1.1.7/1_1_7.sce b/26/CH1/EX1.1.7/1_1_7.sce
new file mode 100755
index 000000000..b4fb8a935
--- /dev/null
+++ b/26/CH1/EX1.1.7/1_1_7.sce
@@ -0,0 +1,8 @@
+disp('the augmented matrix is:')

+a=[1 7 3 -4;0 1 -1 3;0 0 0 1;0 0 1 -2]

+disp(a)

+disp('interchange R3 and R4')

+a([3,4],:)=a([4,3],:)

+disp(a)

+disp('from R4 we get 0=1')

+disp('hence, no solution')
\ No newline at end of file
diff --git a/26/CH1/EX1.2.13/1_2_13.sce b/26/CH1/EX1.2.13/1_2_13.sce
new file mode 100755
index 000000000..bf3f43560
--- /dev/null
+++ b/26/CH1/EX1.2.13/1_2_13.sce
@@ -0,0 +1,14 @@
+disp('the augmented matrix is')

+a=[1 -3 0 -1 0 -2;0 1 0 0 -4 1;0 0 0 1 9 4;0 0 0 0 0 0]

+disp(a)

+disp('R1=R1+R3')

+a(1,:)=a(1,:)+a(3,:)

+disp(a)

+disp('R1=R1+3*R2')

+a(1,:)=a(1,:)+3*a(2,:)

+disp(a)

+disp('corresponding equations are:')

+disp('x1-3*x5=5, x2-4*x5=1, x4+9*x5=4, and 0=0')

+disp('free variables:x3, x5')

+disp('general solution is:')

+disp('x1=5+3*x5, x2=1+4*x5, x3, x4=4-9*x5, x5')
\ No newline at end of file
diff --git a/26/CH1/EX1.2.34/1_2_34.sce b/26/CH1/EX1.2.34/1_2_34.sce
new file mode 100755
index 000000000..cd7923dc5
--- /dev/null
+++ b/26/CH1/EX1.2.34/1_2_34.sce
@@ -0,0 +1,48 @@
+disp('the augmented matrix is:')

+a=[1 0 0 0 0 0 0;1 2 4 8 16 32 2.9;1 4 16 64 256 1024 14.8;1 6 36 216 1296 7776 39.6;1 8 64 512 4096 32768 74.3;1 10 10^2 10^3 10^4 10^5 119];

+disp(a)

+disp('performing row transformations')

+for k=2:6

+  a(k,:)=a(k,:)-a(1,:)

+end

+disp(a)

+j=2;

+for k=3:6

+  a(k,:)=a(k,:)-j*a(2,:)

+  j=j+1;

+end

+disp(a)

+j=[0 0 0 3 6 10];

+for k=4:6

+  a(k,:)=a(k,:)-j(k)*a(3,:)

+end

+disp(a)

+a(5,:)=a(5,:)-4*a(4,:)

+a(6,:)=a(6,:)-10*a(4,:)

+disp(a)

+a(6,:)=a(6,:)-5*a(5,:)

+disp(a)

+a(6,:)=a(6,:)/a(6,6)

+disp(a)

+j=[0 32 960 4800 7680]

+for k=1:5

+  a(k,:)=a(k,:)-j(k)*a(6,:)

+end

+disp(a)

+a(5,:)=a(5,:)/a(5,5)

+j=[0 16 224 576]

+for k=2:4

+  a(k,:)=a(k,:)-j(k)*a(5,:)

+end

+a(4,:)=a(4,:)/48

+a(2,:)=a(2,:)-8*a(4,:)

+a(3,:)=a(3,:)-48*a(4,:)

+a(3,:)=a(3,:)/8

+a(2,:)=a(2,:)-4*a(3,:)

+a(2,:)=a(2,:)/2

+disp(a)

+v=[a(1,7) a(2,7) a(3,7) a(4,7) a(5,7) a(6,7)]

+p=poly(v,"t","coeff")

+disp('p(t)=')

+disp(p)

+disp('p(7.5)=64.6 hundred lb')
\ No newline at end of file
diff --git a/26/CH1/EX1.2.7/1_2_7.sce b/26/CH1/EX1.2.7/1_2_7.sce
new file mode 100755
index 000000000..fd471e1b4
--- /dev/null
+++ b/26/CH1/EX1.2.7/1_2_7.sce
@@ -0,0 +1,18 @@
+disp('the augmented matrix is')

+a=[1 3 4 7;3 9 7 6]

+disp(a)

+disp('R2=R2-3*R1')

+a(2,:)=a(2,:)-3*a(1,:)

+disp(a)

+disp('(-1/5)*R2')

+a(2,:)=(-1/5)*a(2,:)

+disp(a)

+disp('R1=R1-4*R2')

+a(1,:)=a(1,:)-4*a(2,:)

+disp('the row reduced form is:')

+disp(a)

+disp('corresponding equations are')

+disp('x1+3*x2=-5 and x3=3')

+disp('free variables:x2')

+disp('general solution is:')

+disp('x1=-5-3*x2, x2, x3=3')
\ No newline at end of file
diff --git a/26/CH1/EX1.3.1/1_3_1.sce b/26/CH1/EX1.3.1/1_3_1.sce
new file mode 100755
index 000000000..ea1439d38
--- /dev/null
+++ b/26/CH1/EX1.3.1/1_3_1.sce
@@ -0,0 +1,9 @@
+u=[-1;2]

+disp('u=')

+disp(u)

+v=[-3;-1]

+disp('v=')

+disp(v)

+s=u-2*v

+disp('u-2v=')

+disp(s)
\ No newline at end of file
diff --git a/26/CH1/EX1.3.11/1_3_11.sce b/26/CH1/EX1.3.11/1_3_11.sce
new file mode 100755
index 000000000..10b4d51a6
--- /dev/null
+++ b/26/CH1/EX1.3.11/1_3_11.sce
@@ -0,0 +1,17 @@
+disp('vectors a1 a2 a3 are:')

+a1=[1 0 -2]

+disp(a1')

+a2=[-4 3 8]

+disp(a2')

+a3=[2 5 -4]

+disp(a3')

+disp('vector b=')

+b=[3 -7 -3]

+disp(b')

+disp('the augmented matrix is:')

+a=[1 -4 2 3;0 3 5 -7;-2 8 -4 -3]

+disp(a)

+a(3,:)=a(3,:)+2*a(1,:)

+disp(a)

+disp('from the entries of last row, the system is inconsistent')

+disp('hence, b is not a linear combination of a1 a2 and a3')
\ No newline at end of file
diff --git a/26/CH1/EX1.3.31/1_3_31.sce b/26/CH1/EX1.3.31/1_3_31.sce
new file mode 100755
index 000000000..0f6c48ba8
--- /dev/null
+++ b/26/CH1/EX1.3.31/1_3_31.sce
@@ -0,0 +1,32 @@
+disp('1 gram at (0,1), 1 gram at (8,1) and 1 gram at (2,4)')

+cm=(1/3)*(1*[0;1]+1*[8;1]+1*[2;4])

+disp('centre of mass is at')

+disp(cm)

+disp('the new weight of the system=9 grams')

+disp('new centre of mass is at')

+s=[2;2]

+disp(s)

+disp('let w1,w2 and w3 be the weights added at (0,1),(8,1) and (2,4) respectively')

+disp('hence, w1+w2+w3=6')

+disp('using the formula for the centre of mass, we get')

+disp('8*w2+2*w3=8 and w1+w2+4*w3=12')

+a=[1 1 1 6;0 8 2 8;1 1 4 12]

+disp('the augmented matrix is:')

+disp(a)

+disp('R3=R3-R1')

+a(3,:)=a(3,:)-a(1,:)

+disp(a)

+disp('R3=(1/3)*R3')

+a(3,:)=(1/3)*a(3,:)

+disp(a)

+disp('R2=R2-2*R3 and R1=R1-R3')

+a(2,:)=a(2,:)-2*a(3,:)

+a(1,:)=a(1,:)-a(3,:)

+disp(a)

+disp('R1=R1-(1/8)*R2')

+a(1,:)=a(1,:)-(1/8)*a(2,:)

+disp(a)

+disp('R2=(1/8)*R2')

+a(2,:)=(1/8)*a(2,:)

+disp(a)

+printf('Add %.1f grams at (0,1), %.1f grams at (8,1) and %d grams at (2,4)',a(1,4),a(2,4),a(3,4)) 
\ No newline at end of file
diff --git a/26/CH1/EX1.4.13/1_4_13.sce b/26/CH1/EX1.4.13/1_4_13.sce
new file mode 100755
index 000000000..1b54d3d16
--- /dev/null
+++ b/26/CH1/EX1.4.13/1_4_13.sce
@@ -0,0 +1,15 @@
+disp('the augmented matrix is:')

+a=[3 -5 0;-2 6 4;1 1 4]

+disp(a)

+disp('interchange R1 and R3')

+a([1,3],:)=a([3,1],:)

+disp(a)

+disp('R2=R2+2*R1 and R3=R3-3*R1')

+a(2,:)=a(2,:)+2*a(1,:)

+a(3,:)=a(3,:)-3*a(1,:)

+disp(a)

+disp('R3=R3+R2')

+a(3,:)=a(3,:)+a(2,:)

+disp(a)

+disp('from the entries of last row, the system is consistent')

+disp('hence, u is in the plane spanned by the columns of a')
\ No newline at end of file
diff --git a/26/CH1/EX1.4.7/1_4_7.sce b/26/CH1/EX1.4.7/1_4_7.sce
new file mode 100755
index 000000000..b9614b2e4
--- /dev/null
+++ b/26/CH1/EX1.4.7/1_4_7.sce
@@ -0,0 +1,11 @@
+disp('the three vectors are:')

+u=[4;-1;7;-4]

+v=[-5;3;-5;1]

+w=[7;-8;0;2]

+disp(w,v,u)

+disp('u v and w form the columns of A')

+A=[u v w]

+disp(A)

+disp('the augmented matrix is:')

+c=[A [6 -8 0 -7]']

+disp(c)
\ No newline at end of file
diff --git a/26/CH1/EX1.5.1/1_5_1.sce b/26/CH1/EX1.5.1/1_5_1.sce
new file mode 100755
index 000000000..943043ee1
--- /dev/null
+++ b/26/CH1/EX1.5.1/1_5_1.sce
@@ -0,0 +1,12 @@
+disp('the augmented matrix is:')

+a=[2 -5 8 0;-2 -7 1 0;4 2 7 0]

+disp(a)

+disp('R2=R2+2*R1 and R3=R3-2*R1')

+a(2,:)=a(2,:)+a(1,:)

+a(3,:)=a(3,:)-2*a(1,:)

+disp(a)

+disp('R3=R3+R2')

+a(3,:)=a(3,:)+a(2,:)

+disp(a)

+disp('only two columns have non zero pivots')

+disp('hence, one column is a free column and therefore there exists a non trivial solution')
\ No newline at end of file
diff --git a/26/CH1/EX1.5.11/1_5_11.sce b/26/CH1/EX1.5.11/1_5_11.sce
new file mode 100755
index 000000000..5c8e0e0d2
--- /dev/null
+++ b/26/CH1/EX1.5.11/1_5_11.sce
@@ -0,0 +1,13 @@
+disp('the augmented matrix is')

+a=[1 -4 -2 0 3 -5 0;0 0 1 0 0 -1 0;0 0 0 0 -1 4 0;0 0 0 0 0 0 0]

+disp(a)

+disp('R1=R1-3*R3')

+a(1,:)=a(1,:)-3*a(3,:)

+disp(a)

+disp('R1=R1+2*R2')

+a(1,:)=a(1,:)+2*a(2,:)

+disp(a)

+disp('the free variables are:x2, x4 and x6')

+disp('the basic variables are:x1, x3 and x5')

+disp('the solution is:')

+disp('[4*x2-5*x6  x2  x6  x4  4*x6  x6]')
\ No newline at end of file
diff --git a/26/CH1/EX1.5.7/1_5_7.sce b/26/CH1/EX1.5.7/1_5_7.sce
new file mode 100755
index 000000000..557147482
--- /dev/null
+++ b/26/CH1/EX1.5.7/1_5_7.sce
@@ -0,0 +1,12 @@
+disp('the augmented matrix is:')

+a=[1 3 -3 7 0;0 1 -4 5 0]

+disp(a)

+disp('R1=R1-3*R2')

+a(1,:)=a(1,:)-3*a(2,:)

+disp(a)

+disp('basic variables:x1 x2')

+disp('free variables:x3 x4')

+disp('x1=-9*x3+8*x4')

+disp('x2=4*x3-5*x4')

+disp('hence, solution is')

+disp('[-9*x3+8*x4 4*x3-5*x4 x3 x4]')
\ No newline at end of file
diff --git a/26/CH1/EX1.7.1/1_7_1.sce b/26/CH1/EX1.7.1/1_7_1.sce
new file mode 100755
index 000000000..070441bb7
--- /dev/null
+++ b/26/CH1/EX1.7.1/1_7_1.sce
@@ -0,0 +1,15 @@
+disp('given vectors u, v and w are')

+u=[5 0 0]'

+disp(u)

+v=[7 2 -6]'

+disp(v)

+w=[9 4 -8]'

+disp(w)

+disp('the augmented matrix is')

+a=[5 7 9 0;0 2 4 0;0 -6 -8 0]

+disp(a)

+disp('R3=R3+3*R2')

+a(3,:)=a(3,:)+3*a(2,:)

+disp(a)

+disp('there are no free variables')

+disp('hence, the homogeneous equation has only trivial solution and  the vectors are linearly independent')
\ No newline at end of file
diff --git a/26/CH1/EX1.7.7/1_7_7.sce b/26/CH1/EX1.7.7/1_7_7.sce
new file mode 100755
index 000000000..92666e359
--- /dev/null
+++ b/26/CH1/EX1.7.7/1_7_7.sce
@@ -0,0 +1,6 @@
+disp('the augmented matrix is')

+A=[1 -3 3 -2 0;-3 7 -1 2 0;-4 -5 7  5 0]

+disp(A)

+disp('since there are three rows, the maximum number of pivots can be 3')

+disp('hence, at least one of the four variable must be free')

+disp('so the equations have non trivial solution and the columns of A are linearly independent')
\ No newline at end of file
diff --git a/26/CH2/EX2.1.1/2_1_1.sce b/26/CH2/EX2.1.1/2_1_1.sce
new file mode 100755
index 000000000..31c397c92
--- /dev/null
+++ b/26/CH2/EX2.1.1/2_1_1.sce
@@ -0,0 +1,10 @@
+A=[2 0 -1;4 -5 2];

+disp('matrix A:')

+disp(A)

+disp('-2A=')

+disp(-2*A)

+disp('matrix B')

+B=[7 -5 1;1 -4 -3];

+disp(B)

+disp('B-2A=')

+disp(B-2*A)
\ No newline at end of file
diff --git a/26/CH2/EX2.2.1/2_2_1.sce b/26/CH2/EX2.2.1/2_2_1.sce
new file mode 100755
index 000000000..418dcc238
--- /dev/null
+++ b/26/CH2/EX2.2.1/2_2_1.sce
@@ -0,0 +1,5 @@
+disp('given matrix:')

+a=[8 6;5 4];

+disp(a)

+disp('inverse of the matrix is:')

+disp(inv(a))
\ No newline at end of file
diff --git a/26/CH2/EX2.2.7/2_2_7.sce b/26/CH2/EX2.2.7/2_2_7.sce
new file mode 100755
index 000000000..3df602ad4
--- /dev/null
+++ b/26/CH2/EX2.2.7/2_2_7.sce
@@ -0,0 +1,9 @@
+disp('the co-efficient matrix is:')

+a=[1 2;5 12]

+disp(a)

+disp('inverse of the matrix is:')

+disp(inv(a))

+disp('solution is:')

+b=[-1;3];

+c=inv(a);

+disp(c*b)
\ No newline at end of file
diff --git a/26/CH2/EX2.3.1/2_3_1.sce b/26/CH2/EX2.3.1/2_3_1.sce
new file mode 100755
index 000000000..5b3c61e9e
--- /dev/null
+++ b/26/CH2/EX2.3.1/2_3_1.sce
@@ -0,0 +1,6 @@
+disp('the given matrix is:')

+a=[5 7;-3 -6];

+disp(a)

+disp('the columns are lineraly independent')

+disp('hence, by invertible matrix theorem')

+disp('the matrix A is invertible')
\ No newline at end of file
diff --git a/26/CH2/EX2.3.33/2_3_33.sce b/26/CH2/EX2.3.33/2_3_33.sce
new file mode 100755
index 000000000..c09bf4b98
--- /dev/null
+++ b/26/CH2/EX2.3.33/2_3_33.sce
@@ -0,0 +1,10 @@
+disp('matrix A corresponding to transformation T is:')

+A=[-5 9;4 -7];

+disp(A)

+disp('determinant of A is:')

+disp(det(A))

+disp('since det(A) is not equal to zero')

+disp('by IMT, A is invertible')

+disp('hence, the inverse of A exists')

+disp('inverse of A is:')

+disp(inv(A))
\ No newline at end of file
diff --git a/26/CH2/EX2.4.25/2_4_25.sce b/26/CH2/EX2.4.25/2_4_25.sce
new file mode 100755
index 000000000..985a344eb
--- /dev/null
+++ b/26/CH2/EX2.4.25/2_4_25.sce
@@ -0,0 +1,29 @@
+disp('given matrix is:')

+a=[1 2 0 0 0;3 5 0 0 0;0 0 2 0 0;0 0 0 7 8;0 0 0 5 6];

+disp(a)

+disp('partitioning the matrix into 4 submatrices')

+A11=[a(1,1:2);a(2,1:2)]

+disp(A11,'A11=')

+A22=[a(3,3:5);a(4,3:5);a(5,3:5)]

+disp(A22,'A22=')

+A12=zeros(2,3)

+disp(A12,'A12=')

+A21=zeros(3,2)

+disp(A21,'A21=')

+disp('partitioning A22 into 4 submatrices')

+A221=[2]

+disp(A221)

+B=[A22(2,2:3);A22(3,2:3)]

+disp(B,'B=')

+disp(zeros(1,2))

+disp(zeros(2,1))

+disp('determinant of B=')

+disp(det(B))

+disp('Hence, B is invertible')

+disp('inverse of B is')

+disp(inv(B))

+disp('determinant of inverse of  B is:')

+disp(det(inv(B)))

+disp('hence the invese of A22 is:')

+c=[det(inv(B)) zeros(1,2);0 3 -4;0 -2.5 3.5];

+disp(c)
\ No newline at end of file
diff --git a/26/CH2/EX2.5.1/2_5_1.sce b/26/CH2/EX2.5.1/2_5_1.sce
new file mode 100755
index 000000000..591f40ab1
--- /dev/null
+++ b/26/CH2/EX2.5.1/2_5_1.sce
@@ -0,0 +1,45 @@
+disp('the lower triangular matrix is:')

+L=[1 0 0;-1 1 0;2 -5 1];

+disp(L)

+disp('the upper triangular matrix is:')

+U=[3 -7 -2;0 -2 -1;0 0 -1];

+disp(U)

+disp('the RHS of the equations are')

+b=[-7;5;2];

+disp(b)

+disp('combining matrices L and b')

+c=[L b];

+disp(c)

+disp('performing row operations')

+disp('R2=R2+R1')

+c(2,:)=c(2,:)+c(1,:)

+disp(c)

+disp('R3=R3-2*R1')

+c(3,:)=c(3,:)-2*c(1,:)

+disp(c)

+disp('R3=R3+5*R2')

+c(3,:)=c(3,:)+5*c(2,:)

+disp(c)

+y=c(:,4)

+disp(y,'y=')

+disp('combining U and y')

+d=[U y];

+disp(d)

+disp('performing row operations')

+disp('R3=R3/-6')

+d(3,:)=d(3,:)/(-1)

+disp(d)

+disp('R2=R2+R3 and R1=R1+2*R3')

+d(2,:)=d(2,:)+d(3,:)

+d(1,:)=d(1,:)+2*d(3,:)

+disp(d)

+disp('R1=R1-3.5*R2')

+d(1,:)=d(1,:)-3.5*d(2,:)

+disp(d)

+disp('R1=R1/3 and R2=R2/-2')

+d(1,:)=d(1,:)/3

+d(2,:)=d(2,:)/(-2)

+disp(d)

+disp('the solution is:')

+x=d(:,4)

+disp(x,'x=')
\ No newline at end of file
diff --git a/26/CH2/EX2.5.13/2_5_13.sce b/26/CH2/EX2.5.13/2_5_13.sce
new file mode 100755
index 000000000..9f35269eb
--- /dev/null
+++ b/26/CH2/EX2.5.13/2_5_13.sce
@@ -0,0 +1,19 @@
+disp('given matrix is:')

+a=[1 3 -5 -3;-1 -5 8 4;4 2 -5 -7;-2 -4 7 5]

+d=a;

+disp(a)

+disp('performing row operations')

+p21=a(2,1)/a(1,1);p31=a(3,1)/a(1,1);p41=a(4,1)/a(1,1);

+a(2,:)=a(2,:)-p21*a(1,:)

+a(3,:)=a(3,:)-p31*a(1,:)

+a(4,:)=a(4,:)-p41*a(1,:)

+disp(a)

+p32=a(3,2)/a(2,2);p42=a(4,2)/a(2,2)

+a(3,:)=a(3,:)-p32*a(2,:)

+a(4,:)=a(4,:)-p42*a(2,:)

+disp(a)

+disp('thus, lower triangular matrix is:')

+L=[1 0 0 0;p21 1 0 0;p31 p32 1 0;p41 p42 0 1]

+disp(L,'L=')

+disp('Upper triangular matrix is:')

+disp(a,'U=')
\ No newline at end of file
diff --git a/26/CH2/EX2.5.7/2_5_7.sce b/26/CH2/EX2.5.7/2_5_7.sce
new file mode 100755
index 000000000..22c2cd9ee
--- /dev/null
+++ b/26/CH2/EX2.5.7/2_5_7.sce
@@ -0,0 +1,14 @@
+disp('given matrix is:')

+a=[2 5;-3 -4]

+d=a;

+disp(a)

+disp('performing row operations')

+a(2,:)=a(2,:)-(a(2,1)/a(1,1))*a(1,:)

+disp(a)

+disp(a)

+disp('thus, the upper triangular matrix is')

+U=a;

+disp(U,'U=')

+disp('the lower triangular matrix is:')

+L=[1 0;d(2,1)/d(1,1) 1];

+disp(L,'L=')
\ No newline at end of file
diff --git a/26/CH2/EX2.6.1/2_6_1.sce b/26/CH2/EX2.6.1/2_6_1.sce
new file mode 100755
index 000000000..206e97991
--- /dev/null
+++ b/26/CH2/EX2.6.1/2_6_1.sce
@@ -0,0 +1,9 @@
+disp('the consumption matrix is:')

+C=[.1 .6 .6;.3 .2 0;.3 .1 .1];

+disp(C)

+disp('Assuming that agriculture plans to produce 100 units and other units produce nothing')

+disp('the production vector is given by')

+x=[0;100;0];

+disp(x,'x=')

+disp('thus the intermediate demand is:')

+disp(C*x)
\ No newline at end of file
diff --git a/26/CH2/EX2.6.7/2_6_7.sce b/26/CH2/EX2.6.7/2_6_7.sce
new file mode 100755
index 000000000..39cc4b315
--- /dev/null
+++ b/26/CH2/EX2.6.7/2_6_7.sce
@@ -0,0 +1,15 @@
+disp('the consumption matrix is:')

+C=[0 .5;.6 .2];

+disp(C)

+disp('the demand for 1 unit of output sector 1')

+d1=[1;0]

+disp(d1)

+disp('the production required to satisfy demand d1 is:')

+x1=inv(eye(2,2)-C)*d1

+disp(x1,'x1=')

+disp('the final demand is:')

+d2=[51;30]

+disp(d2,'d2=')

+disp('the production required to satisfy demand d2 is:')

+x2=inv(eye(2,2)-C)*d2

+disp(x2,'x2=')
\ No newline at end of file
diff --git a/26/CH2/EX2.7.1/2_7_1.sce b/26/CH2/EX2.7.1/2_7_1.sce
new file mode 100755
index 000000000..2c2dd056b
--- /dev/null
+++ b/26/CH2/EX2.7.1/2_7_1.sce
@@ -0,0 +1,17 @@
+disp('consider the matrix')

+a=[1 .25 0;0 1 0;0 0 1]

+disp(a)

+disp('consider a vector')

+x=[6;8;0]

+disp(x)

+disp('the effect of the matric on the vector is:')

+disp(a*x)

+disp('now consider the matrix:')

+b=[1 .25;0 1]

+disp(b)

+disp('considering the same vector')

+x1=[6;8]

+disp(x1)

+disp('the effect of the new matrix on the vector is:')

+disp(b*x1)

+disp('thus we can see that the two matrices have the same effect on vectors')
\ No newline at end of file
diff --git a/26/CH2/EX2.7.7/2_7_7.sce b/26/CH2/EX2.7.7/2_7_7.sce
new file mode 100755
index 000000000..2fe440582
--- /dev/null
+++ b/26/CH2/EX2.7.7/2_7_7.sce
@@ -0,0 +1,11 @@
+disp('the matrix in R2 to rotate a vector by 60 degrees is:')

+a=[cos(%pi/3) -sin(%pi/3);sin(%pi/3) cos(%pi/3)]

+disp(a)

+x=[6;8]

+disp(x,'x=')

+disp('so the 3X3 matrix for rotation about x is:')

+y=[1 0 6;0 1 8;0 0 1]

+z=[1 0 -6;0 1 -8;0 0 1]

+a=[cos(%pi/3) -sin(%pi/3) 0;sin(%pi/3) cos(%pi/3) 0;0 0 1]

+R=y*(a*z)

+disp(R)
\ No newline at end of file
diff --git a/26/CH2/EX2.8.23/2_8_23.sce b/26/CH2/EX2.8.23/2_8_23.sce
new file mode 100755
index 000000000..14c68ea25
--- /dev/null
+++ b/26/CH2/EX2.8.23/2_8_23.sce
@@ -0,0 +1,20 @@
+disp('the given matrix is:')

+a=[4 5 9 -2;6 5 1 12;3 4 8 -3]

+disp(a)

+disp('performing row operaions')

+a(2,:)=a(2,:)-(a(2,1)/a(1,1))*a(1,:)

+a(3,:)=a(3,:)-(a(3,1)/a(1,1))*a(1,:)

+disp(a)

+a(3,:)=a(3,:)-(a(3,2)/a(2,2))*a(2,:)

+disp(a)

+a(1,:)=a(1,:)/a(1,1)

+a(2,:)=a(2,:)/a(2,2)

+disp(a)

+for i=1:3

+  for j=i:4

+    if(a(i,j)<>0)

+       disp('is a pivot column',j,'column')

+       break

+    end

+  end

+ end
\ No newline at end of file
diff --git a/26/CH2/EX2.8.25/2_8_25.sce b/26/CH2/EX2.8.25/2_8_25.sce
new file mode 100755
index 000000000..f66b24d21
--- /dev/null
+++ b/26/CH2/EX2.8.25/2_8_25.sce
@@ -0,0 +1,21 @@
+disp('the given matrix is:')

+a=[1 4 8 -3 -7;-1 2 7 3 4;-2 2 9 5 5;3 6 9 -5 -2]

+disp(a)

+disp('performing row operations')

+a(2,:)=a(2,:)-(a(2,1)/a(1,1))*a(1,:)

+a(3,:)=a(3,:)-(a(3,1)/a(1,1))*a(1,:)

+a(4,:)=a(4,:)-(a(4,1)/a(1,1))*a(1,:)

+disp(a)

+a(3,:)=a(3,:)-(a(3,2)/a(2,2))*a(2,:)

+a(4,:)=a(4,:)-(a(4,2)/a(2,2))*a(2,:)

+disp(a)

+a(4,:)=a(4,:)-(a(4,4)/a(3,4))*a(3,:)

+disp(a)

+for i=1:4

+  for j=i:5

+    if(a(i,j)<>0)

+      disp('is a pivot column',j,'column')

+      break

+    end

+  end

+end
\ No newline at end of file
diff --git a/26/CH2/EX2.8.7/2_8_7.sce b/26/CH2/EX2.8.7/2_8_7.sce
new file mode 100755
index 000000000..b764b5efb
--- /dev/null
+++ b/26/CH2/EX2.8.7/2_8_7.sce
@@ -0,0 +1,20 @@
+disp('the given matrix is:')

+A=[2 -3 -4;-8 8 6;6 -7 -7]

+disp(A,'A=')

+disp('the given vector is:')

+p=[6;-10;11]

+disp(p,'p=')

+disp('combining A and p')

+b=[A p]

+disp(b)

+disp('performing row operations')

+b(2,:)=b(2,:)-(b(2,1)/b(1,1))*b(1,:)

+b(3,:)=b(3,:)-(b(3,1)/b(1,1))*b(1,:)

+disp(b)

+b(3,:)=b(3,:)-(b(3,2)/b(2,2))*b(2,:)

+disp(b)

+if(b(3,3)==0 & b(3,4)==0)

+    disp('p lies in column space of A')

+  else

+    disp('p does not lie in column space of A')

+ end
\ No newline at end of file
diff --git a/26/CH2/EX2.9.13/2_9_13.sce b/26/CH2/EX2.9.13/2_9_13.sce
new file mode 100755
index 000000000..4e0ec53f2
--- /dev/null
+++ b/26/CH2/EX2.9.13/2_9_13.sce
@@ -0,0 +1,22 @@
+disp('the given matrix is:')

+a=[1 -3 2 -4;-3 9 -1 5;2 -6 4 -3;-4 12 2 7]

+disp(a)

+disp('performing row operations')

+a(2,:)=a(2,:)-(a(2,1)/a(1,1))*a(1,:)

+a(3,:)=a(3,:)-(a(3,1)/a(1,1))*a(1,:)

+a(4,:)=a(4,:)-(a(4,1)/a(1,1))*a(1,:)

+disp(a)

+a(4,:)=a(4,:)-2*a(2,:)

+disp(a)

+a(4,:)=a(4,:)-a(3,:)

+disp(a)

+k=0

+for i=1:4

+  for j=i:4

+    if(a(i,j)<>0)

+      k=k+1

+      break

+    end

+  end

+end

+disp(k,'dimension of the matrix=')
\ No newline at end of file
diff --git a/26/CH3/EX3.1.1/3_1_1.sce b/26/CH3/EX3.1.1/3_1_1.sce
new file mode 100755
index 000000000..8d6475612
--- /dev/null
+++ b/26/CH3/EX3.1.1/3_1_1.sce
@@ -0,0 +1,6 @@
+disp('the given matrix is:')

+A=[3 0 4;2 3 2;0 5 -1]

+disp(A)

+disp('calculating det(A) using cofactor expression along first row')

+disp('det(A)=3 X (-1 X 3-5 X 2)+4 X (2 X 5-3 X 0)')

+disp(det(A),'=')
\ No newline at end of file
diff --git a/26/CH3/EX3.1.13/3_1_13.sce b/26/CH3/EX3.1.13/3_1_13.sce
new file mode 100755
index 000000000..13e9d1d84
--- /dev/null
+++ b/26/CH3/EX3.1.13/3_1_13.sce
@@ -0,0 +1,18 @@
+disp('the given matrix is:')

+A=[4 0 -7 3 -5;0 0 2 0 0;7 3 -6 4 -8;5 0 5 2 -3;0 0 9 -1 2]

+disp(A,'A=')

+P=A

+disp('since row 2 has maximum zeros, using row 2 for cofactor expression')

+A(2,:)=[]

+A(:,3)=[]

+disp('deleting second row and third column from A, we get')

+disp(A)

+disp(A,'det','det(A)=-2 X')

+disp('for the 4X4 matrix obtained, using column 2 for cofactor exansion')

+disp('deleting second column and row from the 4X4 matrix')

+A(2,:)=[]

+A(:,2)=[]

+disp(A)

+disp(A,'det','det(A)=-2 X 3 X')

+disp('-6 X [4 X (4-3)-5 X (6-5)]','=')

+disp(-6*det(A),'=')
\ No newline at end of file
diff --git a/26/CH3/EX3.1.19/3_1_19.sce b/26/CH3/EX3.1.19/3_1_19.sce
new file mode 100755
index 000000000..452a8e568
--- /dev/null
+++ b/26/CH3/EX3.1.19/3_1_19.sce
@@ -0,0 +1,14 @@
+disp('the given matrix is:')

+disp('A=')

+disp('a  b')

+disp('c  d')

+disp('det(A)=ad-bc')

+disp('interchanging the rows of A, we get')

+disp('B=')

+disp('c  d')

+disp('a  b')

+disp('det(B)=bc-ad')

+disp('-(ad-bc)','=')

+disp('-det(A)','=')

+disp('interchanging 2 rows reverses the sign of the determinant')

+disp('at least for the 2X2 case')
\ No newline at end of file
diff --git a/26/CH3/EX3.1.37/3_1_37.sce b/26/CH3/EX3.1.37/3_1_37.sce
new file mode 100755
index 000000000..a101d43ef
--- /dev/null
+++ b/26/CH3/EX3.1.37/3_1_37.sce
@@ -0,0 +1,10 @@
+A=[3 1;4 2]

+disp('the given matrix is:')

+disp(A)

+disp(det(A),'det(A)=')

+disp('5 X A = ')

+disp(5*A)

+disp(det(5*A),'det(5*A)=')

+disp('thus, det(5A) is not equal to 5Xdet(A)')

+disp('infact, the relation between det(rA) and det(A) for a nxn matrix is:')

+disp('det(rA)=(r^n)*det(A)')
\ No newline at end of file
diff --git a/26/CH3/EX3.1.7/3_1_7.sce b/26/CH3/EX3.1.7/3_1_7.sce
new file mode 100755
index 000000000..d33844f04
--- /dev/null
+++ b/26/CH3/EX3.1.7/3_1_7.sce
@@ -0,0 +1,6 @@
+disp('given matrix is:')

+A=[4 3 0;6 5 2;9 7 3]

+disp(A)

+disp('calculating det(A) using cofactor expression along first row')

+disp('det(A)=4 X (5 X 3-7 X 2)-3 X (6 X 3-9 X 2)')

+disp(det(A),'=')
\ No newline at end of file
diff --git a/26/CH3/EX3.2.13/3_2_13.sce b/26/CH3/EX3.2.13/3_2_13.sce
new file mode 100755
index 000000000..e879e9b4a
--- /dev/null
+++ b/26/CH3/EX3.2.13/3_2_13.sce
@@ -0,0 +1,18 @@
+disp('the given matrix is:')

+a=[2 5 4 1;4 7 6 2;6 -2 -4 0;-6 7 7 0]

+disp(a,'A=')

+disp('performing row operations')

+a(2,:)=a(2,:)-2*a(1,:)

+disp(a)

+disp('using cofactor expansion about fourth column')

+a(1,:)=[]

+a(:,4)=[]

+disp(a,'det','det(A)= -1 X')

+disp('performing row operations')

+a(3,:)=a(3,:)+a(2,:)

+disp(a)

+disp('using cofactor expansion about first column')

+a(2,:)=[]

+a(:,1)=[]

+disp(a,'det','det(A)= -1 X -6 X')

+disp(6*det(a),'=')
\ No newline at end of file
diff --git a/26/CH3/EX3.2.19/3_2_19.sce b/26/CH3/EX3.2.19/3_2_19.sce
new file mode 100755
index 000000000..60a8c6579
--- /dev/null
+++ b/26/CH3/EX3.2.19/3_2_19.sce
@@ -0,0 +1,25 @@
+disp('the given matrix is:')

+disp('A=')

+disp(' a       b       c')

+disp('2d+a    2e+b    2f+c')

+disp(' g       h       i')

+disp('B=')

+disp('a  b  c')

+disp('d  e  f')

+disp('g  h  i')

+disp('given, det(B)=7')

+disp('performing row operations on A')

+disp('R2=R2-R1')

+disp('A=')

+disp('a   b   c')

+disp('2d  2e  2f')

+disp('g   h   i')

+disp('factoring 2 out of row 2')

+disp('A=')

+disp('2 X')

+disp('a  b  c')

+disp('d  e  f')

+disp('g  h  i')

+disp('therefore, det(A)=2 X det(B)')

+disp('=2 X 7')

+disp('= 14')
\ No newline at end of file
diff --git a/26/CH3/EX3.2.25/3_2_25.sce b/26/CH3/EX3.2.25/3_2_25.sce
new file mode 100755
index 000000000..ef1cc6356
--- /dev/null
+++ b/26/CH3/EX3.2.25/3_2_25.sce
@@ -0,0 +1,13 @@
+disp('the given vectors are:')

+v1=[7 -4 -6]'

+v2=[-8 5 7]'

+v3=[7 0 -5]'

+disp(v3,'v3=',v2,'v2=',v1,'v1=')

+disp('combining them as a matrix')

+a=[v1 v2 v3]

+disp(a,'A=')

+disp('if det(A) is not equal to zero, then v1 v2 and v3 are linearly independent')

+disp('expanding about third column')

+disp('det(A)=7 X (-28+30) - 5 X (35-32)')

+disp(det(a),'=')

+disp('hence, v1 v2 and v3 are linearly independent')
\ No newline at end of file
diff --git a/26/CH3/EX3.2.7/3_2_7.sce b/26/CH3/EX3.2.7/3_2_7.sce
new file mode 100755
index 000000000..08150668d
--- /dev/null
+++ b/26/CH3/EX3.2.7/3_2_7.sce
@@ -0,0 +1,15 @@
+disp('the given matrix is:')

+A=[1 3 0 2;-2 -5 7 4;3 5 2 1;1 -1 2 -3]

+disp(A,'A=')

+disp('performing row operations')

+A(2,:)=A(2,:)-(A(2,1)/A(1,1))*A(1,:)

+A(3,:)=A(3,:)-(A(3,1)/A(1,1))*A(1,:)

+A(4,:)=A(4,:)-(A(4,1)/A(1,1))*A(1,:)

+disp(A)

+A(3,:)=A(3,:)-(A(3,2)/A(2,2))*A(2,:)

+A(4,:)=A(4,:)-(A(4,2)/A(2,2))*A(2,:)

+disp(A)

+A(4,:)=A(4,:)-(A(4,3)/A(3,3))*A(3,:)

+disp(A)

+disp('det(A) is the product of diagonal entries')

+disp(det(A),'det(A)=')
\ No newline at end of file
diff --git a/26/CH3/EX3.3.1/3_3_1.sce b/26/CH3/EX3.3.1/3_3_1.sce
new file mode 100755
index 000000000..70180971b
--- /dev/null
+++ b/26/CH3/EX3.3.1/3_3_1.sce
@@ -0,0 +1,17 @@
+disp('the co-efficient matrix is:')

+a=[5 7;2 4]

+disp(a,'A=')

+disp('the RHS is:')

+b=[3;1]

+disp(b)

+disp('applying cramers rule')

+disp('replacing first column of matrix A by b')

+A1=[3 7;1 4]

+disp(A1,'A1=')

+disp('replacing second column of matrix A by b')

+A2=[5 3;2 1]

+disp(A2,'A2=')

+disp('x1=det(A1)/det(A)')

+disp((det(A1)/det(a)),'=')

+disp('x2=det(A2)/det(A)')

+disp((det(A2)/det(a)),'=')
\ No newline at end of file
diff --git a/26/CH3/EX3.3.13/3_3_13.sce b/26/CH3/EX3.3.13/3_3_13.sce
new file mode 100755
index 000000000..56c60ca2b
--- /dev/null
+++ b/26/CH3/EX3.3.13/3_3_13.sce
@@ -0,0 +1,28 @@
+disp('the given matrix is:')

+a=[3 5 4;1 0 1;2 1 1]

+disp(a,'A=')

+disp('the cofactors are:')

+C11=det([0 1;1 1])

+disp(C11,'C11=')

+C12=-det([1 1;2 1])

+disp(C12,'C12=')

+C13=det([1 0;2 1])

+disp(C13,'C13=')

+C21=-det([5 4;1 1])

+disp(C21,'C21=')

+C22=det([3 4;2 1])

+disp(C22,'C22=')

+C23=-det([3 5;2 1])

+disp(C23,'C23=')

+C31=det([5 4;0 1])

+disp(C31,'C31=')

+C32=-det([3 4;1 1])

+disp(C32,'C32=')

+C33=det([3 5;1 0])

+disp(C33,'C33=')

+B=[C11 C12 C13;C21 C22 C23;C31 C32 C33]'

+disp('adj(A)=')

+disp(B)

+C=B/(det(a))

+disp('inv(A)=')

+disp(C)
\ No newline at end of file
diff --git a/26/CH3/EX3.3.19/3_3_19.sce b/26/CH3/EX3.3.19/3_3_19.sce
new file mode 100755
index 000000000..068e15ed6
--- /dev/null
+++ b/26/CH3/EX3.3.19/3_3_19.sce
@@ -0,0 +1,7 @@
+disp('the points forming the parrallelogram are')

+disp('(0,0),(5,2),(6,4),(11,6)')

+disp('using the vertices adjacent to origin to form a matrix')

+A=[5 6;2 4]

+disp(A,'A=')

+disp('Area of parallelogram = det(A)')

+disp(det(A),'=')
\ No newline at end of file
diff --git a/26/CH4/EX4.1.13/4_1_13.sce b/26/CH4/EX4.1.13/4_1_13.sce
new file mode 100755
index 000000000..8a063beb7
--- /dev/null
+++ b/26/CH4/EX4.1.13/4_1_13.sce
@@ -0,0 +1,24 @@
+disp('the given vectors are:')

+v1=[1;0;-1]

+disp(v1,'v1=')

+v2=[2;1;3]

+disp(v2,'v2=')

+v3=[4;2;6]

+disp(v3,'v3=')

+w=[3;1;2]

+disp(w,'w=')

+disp('It is clear that w is not one of the three vectors in v1,v2 and v3')

+disp('The span of v1,v2 and v3 contains infinitely many vectors.')

+disp('To check if w is in the subspace of v1,v2 and v3,')

+disp('we form an augmented matrix.')

+a=[1 2 4 3;0 1 2 1;-1 3 6 2]

+disp(a)

+disp('performing row operations')

+disp('R3=R3+R1')

+a(3,:)=a(3,:)+a(1,:)

+disp(a)

+disp('R3=R3-5xR2')

+a(3,:)=a(3,:)-5*a(2,:)

+disp(a)

+disp('there is no pivot in the augmented column,')

+disp('hence the vector equation is consistent and w is in span{v1 v2 v3}.')
\ No newline at end of file
diff --git a/26/CH4/EX4.2.1/4_2_1.sce b/26/CH4/EX4.2.1/4_2_1.sce
new file mode 100755
index 000000000..f4295db49
--- /dev/null
+++ b/26/CH4/EX4.2.1/4_2_1.sce
@@ -0,0 +1,10 @@
+disp('the given matrix is:')

+a=[3 -5 -3;6 -2 0;-8 4 1]

+disp(a,'A=')

+disp('the vector x is:')

+x=[1;3;-4]

+disp(x,'x=')

+disp('To check if x is in nullspace of A')

+disp('Ax=')

+disp([0;0;0],'=')

+disp('hence, x is in the null space of A')
\ No newline at end of file
diff --git a/26/CH4/EX4.3.13/4_3_13.sce b/26/CH4/EX4.3.13/4_3_13.sce
new file mode 100755
index 000000000..b9838195d
--- /dev/null
+++ b/26/CH4/EX4.3.13/4_3_13.sce
@@ -0,0 +1,12 @@
+disp('the given matrix is:')

+a=[1 0 6 5;0 2 5 3;0 0 0 0]

+p=a

+disp(a,'A=')

+disp('Reducing A to echelon form')

+disp('R2=R2/2')

+a(2,:)=a(2,:)/2

+disp(a)

+disp('the pivot columns are column 1 and 2 of A')

+disp('hence column space of A is:')

+disp('span')

+disp(a(:,1),'and',a(:,2))
\ No newline at end of file
diff --git a/26/CH4/EX4.4.27/4_4_27.sce b/26/CH4/EX4.4.27/4_4_27.sce
new file mode 100755
index 000000000..e83042db6
--- /dev/null
+++ b/26/CH4/EX4.4.27/4_4_27.sce
@@ -0,0 +1,18 @@
+disp('to check if vectors v1 v2 and v3 are linearly independent')

+v1=[1;0;0;1]

+v2=[3;1;-2;0]

+v3=[0;-1;3;-1]

+disp(v3,'v3=',v2,'v2=',v1,'v1=')

+disp('forming an augmented matrix')

+a=[1 3 0 0;0 1 -1 0;0 -2 3 0;1 0 -1 0]

+disp(a,'A=')

+disp('performing row operations')

+a(4,:)=a(4,:)-a(1,:)

+disp(a)

+a(3,:)=a(3,:)+2*a(2,:)

+a(4,:)=a(4,:)+3*a(2,:)

+disp(a)

+a(4,:)=a(4,:)+4*a(3,:)

+disp(a)

+disp('since the vector equation has only the trivial solution')

+disp('vectors v1 v2 and v3 are linearly independent')
\ No newline at end of file
diff --git a/26/CH4/EX4.4.31a/4_4_31_parta.sce b/26/CH4/EX4.4.31a/4_4_31_parta.sce
new file mode 100755
index 000000000..222a2595d
--- /dev/null
+++ b/26/CH4/EX4.4.31a/4_4_31_parta.sce
@@ -0,0 +1,11 @@
+disp('to check if the polynomials span R3')

+disp('placing the coordinate vectors of the polynomial into the columns of a matrix')

+a=[1 -3 -4 1;-3 5 5 0;5 -7 -6 1]

+disp(a,'A=')

+disp('performing row operations')

+a(2,:)=a(2,:)+3*a(1,:)

+a(3,:)=a(3,:)-5*a(1,:)

+disp(a)

+a(3,:)=a(3,:)+2*a(2,:)

+disp(a)

+disp('the four vectors DO NOT span R3 as there is no pivot in row 3')
\ No newline at end of file
diff --git a/26/CH4/EX4.4.31b/4_4_31_partb.sce b/26/CH4/EX4.4.31b/4_4_31_partb.sce
new file mode 100755
index 000000000..356702012
--- /dev/null
+++ b/26/CH4/EX4.4.31b/4_4_31_partb.sce
@@ -0,0 +1,12 @@
+disp('to check if the polynomials span R3')

+disp('placing the coordinate vectors of the polynomial into the columns of a matrix')

+a=[0 1 -3 2;5 -8 4 -3;1 -2 2 0]

+disp(a,'A=')

+disp('performing row operations')

+a([1 3],:)=a([3 1],:)

+disp(a)

+a(2,:)=a(2,:)-5*a(1,:)

+disp(a)

+a(3,:)=a(3,:)-.5*a(2,:)

+disp(a)

+disp('the four vectors SPAN R3 as there is a pivot in each row')
\ No newline at end of file
diff --git a/26/CH4/EX4.4.7/4_4_7.sce b/26/CH4/EX4.4.7/4_4_7.sce
new file mode 100755
index 000000000..ffc73f74b
--- /dev/null
+++ b/26/CH4/EX4.4.7/4_4_7.sce
@@ -0,0 +1,25 @@
+disp('vector x=')

+x=[8;-9;6]

+disp(x)

+disp('the given basis is:')

+b1=[1;-1;-3]

+b2=[-3;4;9]

+b3=[2;-2;4]

+disp(b1,'b1=')

+disp(b2,'b2=')

+disp(b3,'b3=')

+disp('to solve the vector equation')

+disp('an augmented matrix is formed')

+a=[1 -3 2 8;-1 4 -2 -9;-3 9 4 6]

+disp(a,'A=')

+disp('performing row operations')

+a(2,:)=a(2,:)-(a(2,1)/a(1,1))*a(1,:)

+a(3,:)=a(3,:)-(a(3,1)/a(1,1))*a(1,:)

+disp(a)

+a(3,:)=a(3,:)/a(3,3)

+a(1,:)=a(1,:)-2*a(3,:)

+disp(a)

+a(1,:)=a(1,:)+3*a(2,:)

+disp(a)

+disp('Xb=')

+disp(a(:,4))
\ No newline at end of file
diff --git a/26/CH4/EX4.5.3/problem3.sce b/26/CH4/EX4.5.3/problem3.sce
new file mode 100755
index 000000000..021ff24ea
--- /dev/null
+++ b/26/CH4/EX4.5.3/problem3.sce
@@ -0,0 +1,9 @@
+disp('to find the dimension of subspace H, which is the set of linear combination of vectors v1 v2 and v3')

+v1=[0;1;0;1]

+v2=[0;-1;1;2]

+v3=[2;0;-3;0]

+disp(v3,'v3=',v2,'v2=',v1,'v1=')

+disp('Clearly, v1 is not equal to zero')

+disp('and v2 is not a multiple of v1 as third element of v1 is zero whereas that of v2 is 1.')

+disp('Also, v3 is not a linear combination of v1 and v2 as the first element of v1 and v2 is zero but that of v3 is 2')

+disp('Hence, v1 v2 and v3 are linearly independent and dim(H)=3')
\ No newline at end of file
diff --git a/26/CH4/EX4.6.1/problem1.sce b/26/CH4/EX4.6.1/problem1.sce
new file mode 100755
index 000000000..2fa14bdfb
--- /dev/null
+++ b/26/CH4/EX4.6.1/problem1.sce
@@ -0,0 +1,21 @@
+disp('to find the rank of matrix A')

+a=[1 -4 9 -7;-1 2 -4 1;5 -6 10 7]

+p=a

+disp(a,'A=')

+disp('performing row operations')

+a(2,:)=a(2,:)+a(1,:)

+a(3,:)=a(3,:)-5*a(1,:)

+disp(a)

+a(3,:)=a(3,:)+7*a(2,:)

+disp(a)

+disp('It is clear that matrix A has 2 pivot columns')

+disp('Hence, rank(A)=2')

+disp('COlumns 1 and 2 are pivot columns')

+disp(p(:,1),'and',p(:,2),'Hence, basis for C(A) is:')

+disp('Basis for row space of A is:')

+disp(a(1,:),'and',a(2,:))

+disp('To find the basis of N(A), solve Ax=0')

+disp('on solving, we get the basis of N(A) as:')

+u=[1;2.5;1;0]

+v=[-5;-3;0;1]

+disp(v,'and',u)
\ No newline at end of file
diff --git a/26/CH5/EX5.1.1/5_1_1.sce b/26/CH5/EX5.1.1/5_1_1.sce
new file mode 100755
index 000000000..4f00e19dd
--- /dev/null
+++ b/26/CH5/EX5.1.1/5_1_1.sce
@@ -0,0 +1,8 @@
+disp('to check if 2 is an eigenvalue of matrix A')

+a=[3 2;3 8]

+disp(a,'A=')

+disp('A-2I=')

+b=a-2*eye(2,2)

+disp(b)

+disp('The columns of A are clearly independent,')

+disp('hence (A-2I)x=0 has a non trivial solution and 2 is an eigenvalue of matrix A')
\ No newline at end of file
diff --git a/26/CH5/EX5.1.13/5_1_13.sce b/26/CH5/EX5.1.13/5_1_13.sce
new file mode 100755
index 000000000..0d95a34e0
--- /dev/null
+++ b/26/CH5/EX5.1.13/5_1_13.sce
@@ -0,0 +1,46 @@
+disp('To find a basis for the eigenspace')

+disp('Matrix A=')

+a=[4 0 1;-2 1 0;-2 0 1]

+disp(a)

+disp('for lambda=1')

+disp('A-1I=')

+b=a-eye(3,3)

+disp(b)

+disp('solving (A-I)x=0, we get')

+disp('-2*x1=0 and 3*x1+x3=0')

+disp('therefore, x1=x3=0')

+disp('which leaves x2 as a free variable')

+disp('Hence a basis for the eigen space is:')

+disp([0;1;0])

+disp('for lambda=2')

+disp('A-2I=')

+b=a-2*eye(3,3)

+disp(b)

+disp('performing row operations on the augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+c(2,:)=c(2,:)+c(1,:)

+c(3,:)=c(3,:)+c(1,:)

+disp(c)

+c(1,:)=c(1,:)/c(2,2)

+disp(c)

+disp('We can see that x3 is a free variable')

+disp('x2=x3 and x1=-.05*x3')

+disp('Hence, a basis for the eigenspace is:')

+disp([-.5;1;1])

+disp('for lambda=3')

+disp('A-3I=')

+b=a-3*eye(3,3)

+disp(b)

+disp('performing row operations on the augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+c(2,:)=c(2,:)+2*c(1,:)

+c(3,:)=c(3,:)+2*c(1,:)

+disp(c)

+c(2,:)=c(2,:)/2

+disp(c)

+disp('Again x3 is a free variable')

+disp('x1=-x3 and x2=x3')

+disp('Hence, a basis for the eigenspace is:')

+disp([-1;1;1])
\ No newline at end of file
diff --git a/26/CH5/EX5.1.19/5_1_19.sce b/26/CH5/EX5.1.19/5_1_19.sce
new file mode 100755
index 000000000..3e0f3172c
--- /dev/null
+++ b/26/CH5/EX5.1.19/5_1_19.sce
@@ -0,0 +1,5 @@
+disp('The given matrix is:')

+a=[1 1 1;2 2 2;3 3 3]

+disp(a,'A=')

+disp('A is not invertible because its columns are linearly dependent.')

+disp('Hence, 0 is an eigenvalue of matrix A.')
\ No newline at end of file
diff --git a/26/CH5/EX5.1.7/5_1_7.sce b/26/CH5/EX5.1.7/5_1_7.sce
new file mode 100755
index 000000000..fe6bac4f1
--- /dev/null
+++ b/26/CH5/EX5.1.7/5_1_7.sce
@@ -0,0 +1,22 @@
+disp('To check if 4 is an eigenvalue of matrix A')

+a=[3 0 -1;2 3 1;-3 4 5]

+disp(a,'A=')

+disp('Therefore')

+disp('A-4I=')

+disp(a-4*eye(3,3))

+b=a-4*eye(3,3)

+disp('to check the invertibility of A-4I, form an augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+disp('performing row operations')

+c(2,:)=c(2,:)+2*c(1,:)

+c(3,:)=c(3,:)-3*c(1,:)

+disp(c)

+c(3,:)=c(3,:)+4*c(2,:)

+disp(c)

+disp('We can see that there exists a non trivial solution.')

+disp('Hence, 4 is an eigenvalue of A.')

+disp('For the eigenvector, -x1-x3=0 and -x2-x3=0')

+disp('If x3=1,')

+x=[-1;-1;1]

+disp(x,'x=')
\ No newline at end of file
diff --git a/26/CH5/EX5.2.1/5_2_1.sce b/26/CH5/EX5.2.1/5_2_1.sce
new file mode 100755
index 000000000..658b368b3
--- /dev/null
+++ b/26/CH5/EX5.2.1/5_2_1.sce
@@ -0,0 +1,6 @@
+disp('To find the eigenvalue of matrix A')

+disp('A=')

+a=[2 7;7 2]

+disp(a)

+disp('Eigen values of A are:')

+disp(spec(a))
\ No newline at end of file
diff --git a/26/CH5/EX5.2.13/5_2_13.sce b/26/CH5/EX5.2.13/5_2_13.sce
new file mode 100755
index 000000000..9ceec408d
--- /dev/null
+++ b/26/CH5/EX5.2.13/5_2_13.sce
@@ -0,0 +1,6 @@
+disp('To find the eigenvalues of the matrix A')

+disp('A=')

+a=[6 -2 0;-2 9 0;5 8 3]

+disp(a)

+disp('Eigenvalues of A are:')

+disp(spec(a))
\ No newline at end of file
diff --git a/26/CH5/EX5.2.25/5_2_25.sce b/26/CH5/EX5.2.25/5_2_25.sce
new file mode 100755
index 000000000..a40b91812
--- /dev/null
+++ b/26/CH5/EX5.2.25/5_2_25.sce
@@ -0,0 +1,30 @@
+disp('Matrix A=')

+a=[.6 .3;.4 .7]

+disp(a)

+disp('Eigenvector v1=')

+v1=[3/7;4/7]

+disp(v1)

+disp('vector Xo=')

+Xo=[.5;.5]

+disp(Xo)

+disp('Eigenvalues of A are:')

+c=spec(a)

+disp(c)

+disp('To verify if v1 is an eigenvector of A:')

+disp('A*v1=')

+disp(a*v1)

+disp('=')

+disp('1*v1')

+disp('Hence v1 is an eigenvector of A corresponding to eigenvalue 1.')

+disp('for lambda=.3')

+disp('A-.3I=')

+b=a-.3*eye(2,2)

+disp(b)

+disp('performing row operations on the augmented matrix')

+c=[b [0;0]]

+disp(c)

+c(2,:)=c(2,:)-(c(2,1)/c(1,1))*c(1,:)

+disp(c)

+disp('hence, x1+x2=0')

+disp('Eigenvector corresponding to eigenvalue .3 is:')

+disp([-1;1])
\ No newline at end of file
diff --git a/26/CH5/EX5.2.7/5_2_7.sce b/26/CH5/EX5.2.7/5_2_7.sce
new file mode 100755
index 000000000..22bd3253e
--- /dev/null
+++ b/26/CH5/EX5.2.7/5_2_7.sce
@@ -0,0 +1,7 @@
+disp('To find the eigenvalues of matrix A.')

+disp('A=')

+a=[5 3;-4 4]

+disp(a)

+disp('Eigen values of A are:')

+disp(spec(a))

+disp('Hence, A has no real eigenvalues.')
\ No newline at end of file
diff --git a/26/CH5/EX5.3.1/5_3_1.sce b/26/CH5/EX5.3.1/5_3_1.sce
new file mode 100755
index 000000000..101f3dbc2
--- /dev/null
+++ b/26/CH5/EX5.3.1/5_3_1.sce
@@ -0,0 +1,11 @@
+disp('The given eigenvector matrix is:')

+p=[5 7;2 3]

+disp(p,'P=')

+disp('The diagonal matrix is:')

+d=[2 0;0 1]

+disp(d,'D=')

+disp('Therefore, matrix A=PD(p^-1)')

+s=inv(p)

+disp(p*d*s)

+disp('Hence, A^4=P(D^4)(P^-1)')

+disp(p*(d^4)*s)
\ No newline at end of file
diff --git a/26/CH5/EX5.3.13/5_3_13.sce b/26/CH5/EX5.3.13/5_3_13.sce
new file mode 100755
index 000000000..0168745f3
--- /dev/null
+++ b/26/CH5/EX5.3.13/5_3_13.sce
@@ -0,0 +1,42 @@
+disp('Given matrix A=')

+a=[2 2 -1;1 3 -1;-1 -2 2]

+disp(a)

+disp('Given its eigen values are 5 and 1')

+disp('for lambda=5')

+disp('A-5I=')

+b=a-5*eye(3,3)

+disp(b)

+disp('performing row operations')

+c=[b [0;0;0]]

+disp(c)

+c([1 2],:)=c([2 1],:)

+disp(c)

+c(2,:)=c(2,:)+3*c(1,:)

+c(3,:)=c(3,:)+c(1,:)

+disp(c)

+c(3,:)=c(3,:)-c(2,:)

+disp(c)

+c(2,:)=c(2,:)/c(2,2)

+disp(c)

+disp('With x3 as free variable, x1=-x3 and x2=-x3')

+disp('Hence, for lambda=5 eigenvector is:')

+u1=[-1;-1;1]

+disp(u1)

+disp('for lambda=1')

+disp('A-I=')

+b=a-eye(3,3)

+disp(b)

+disp('performing row operations')

+c=[b [0;0;0]]

+disp(c)

+c(2,:)=c(2,:)-c(1,:)

+c(3,:)=c(3,:)+c(1,:)

+disp(c)

+disp('With x2 and x3 as free variables, eigen vectors corresponding to lambda=1 are')

+u2=[-2;1;0]

+u3=[1;0;1]

+disp(u3,u2)

+disp('Hence, matrix P=')

+disp([u1 u2 u3])

+disp('and matrix D=')

+disp([5 0 0;0 1 0;0 0 1])
\ No newline at end of file
diff --git a/26/CH5/EX5.3.7/5_3_7.sce b/26/CH5/EX5.3.7/5_3_7.sce
new file mode 100755
index 000000000..234d38b9e
--- /dev/null
+++ b/26/CH5/EX5.3.7/5_3_7.sce
@@ -0,0 +1,25 @@
+disp('the given matrix is:')

+a=[1 0;6 -1]

+disp(a,'A=')

+disp('Since A is triangular, eigenvalues are the diagonal entries.')

+disp(a(2,2),a(1,1),'Eigenvalues are:')

+disp('for lambda=1')

+disp('A-1I=')

+b=a-eye(2,2)

+disp(b)

+disp('Hence, x1=(1/3)x2 with x2 as free variable.')

+disp('Eigenvector corresponding to lambda=1 is:')

+u1=[1;3]

+disp(u1)

+disp('for lambda=-1')

+disp('A-(-1)I=')

+b=a+eye(2,2)

+disp(b)

+disp('Hence, x1=0 with x2 as free variable.')

+disp('Eigenvector corresponding to lambda=-1 is:')

+u2=[0;1]

+disp(u2)

+disp('Thus, matrix P=')

+disp([u1 u2])

+disp('and matrix D=')

+disp([1 0;0 -1])
\ No newline at end of file
diff --git a/26/CH5/EX5.4.31/5_4_31.sce b/26/CH5/EX5.4.31/5_4_31.sce
new file mode 100755
index 000000000..121ab0a6b
--- /dev/null
+++ b/26/CH5/EX5.4.31/5_4_31.sce
@@ -0,0 +1,9 @@
+disp('Given matrix A=')

+a=[-7 -48 -16;1 14 6;-3 -45 -19]

+disp(a)

+disp('and matrix P=')

+p=[-3 -2 3;1 1 -1;-3 -3 0]

+disp(p)

+disp('Hence, marix D=')

+s=inv(p)

+disp(s*a*p)
\ No newline at end of file
diff --git a/26/CH5/EX5.5.1/5_5_1.sce b/26/CH5/EX5.5.1/5_5_1.sce
new file mode 100755
index 000000000..e2587b60a
--- /dev/null
+++ b/26/CH5/EX5.5.1/5_5_1.sce
@@ -0,0 +1,16 @@
+disp('Matrix A=')

+a=[1 -2;1 3]

+disp(a)

+disp('Eigen values of A are')

+eig=spec(a)

+disp(eig)

+disp('for lambda=2+i')

+i=sqrt(-1)

+disp('A-(2+i)I=')

+b=a-(2+i)*eye(2,2)

+disp(b)

+disp('With x2 as free variable, x1=-(1-i)x2')

+disp('Hence, eigenvector corresponding to lambda=2+i is:')

+disp([i-1;1])

+disp('for lambda=2-i, eigenvector is:')

+disp([-1-i;1])
\ No newline at end of file
diff --git a/26/CH5/EX5.5.7/5_5_7.sce b/26/CH5/EX5.5.7/5_5_7.sce
new file mode 100755
index 000000000..34c591d9e
--- /dev/null
+++ b/26/CH5/EX5.5.7/5_5_7.sce
@@ -0,0 +1,8 @@
+disp('Matrix A=')

+a=[sqrt(3) -1;1 sqrt(3)]

+disp(a)

+disp('Eigenvalues of A are:')

+eig=spec(a)

+disp(eig)

+disp('The scale factor associated with the transformation x to Ax is:')

+disp(abs(eig(1,1)))
\ No newline at end of file
diff --git a/26/CH6/EX6.1.1/6_1_1.sce b/26/CH6/EX6.1.1/6_1_1.sce
new file mode 100755
index 000000000..b2fee30f0
--- /dev/null
+++ b/26/CH6/EX6.1.1/6_1_1.sce
@@ -0,0 +1,9 @@
+disp('Vectors u an v are:')

+u=[-1;2]

+v=[4;6]

+disp(v,u)

+disp('Projection of v on u=(u.v)/(v.v)')

+a=u'*v

+b=u'*u

+p=a/b

+disp(p,'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.1.13/6_1_13.sce b/26/CH6/EX6.1.13/6_1_13.sce
new file mode 100755
index 000000000..cb26676f7
--- /dev/null
+++ b/26/CH6/EX6.1.13/6_1_13.sce
@@ -0,0 +1,6 @@
+disp('Vector x and y are:')

+x=[10;-3]

+y=[-1;-5]

+disp(y,x)

+disp('||x-y||=sqrt(121+4)')

+disp(sqrt(125),'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.1.7/6_1_7.sce b/26/CH6/EX6.1.7/6_1_7.sce
new file mode 100755
index 000000000..115c7fce5
--- /dev/null
+++ b/26/CH6/EX6.1.7/6_1_7.sce
@@ -0,0 +1,5 @@
+disp('w=')

+w=[3;-1;-5]

+disp(w)

+disp('||w||=sqrt(9+1+25)')

+disp(sqrt(35))
\ No newline at end of file
diff --git a/26/CH6/EX6.2.1/6_2_1.sce b/26/CH6/EX6.2.1/6_2_1.sce
new file mode 100755
index 000000000..fabf3d665
--- /dev/null
+++ b/26/CH6/EX6.2.1/6_2_1.sce
@@ -0,0 +1,10 @@
+disp('To verify if u v and w are orthogonal')

+u=[-1;4;-3]

+v=[5;2;1]

+w=[3;-4;-7]

+disp(w,v,u)

+disp('u.v=')

+disp(v'*u)

+disp('u.w=')

+disp(u'*w)

+disp('Since u.w is not equal to zero, the set {u v w} is not orthogonal.')
\ No newline at end of file
diff --git a/26/CH6/EX6.2.13/6_2_13.sce b/26/CH6/EX6.2.13/6_2_13.sce
new file mode 100755
index 000000000..bdb82fa13
--- /dev/null
+++ b/26/CH6/EX6.2.13/6_2_13.sce
@@ -0,0 +1,11 @@
+disp('Vectors y and u are:')

+y=[2;3]

+u=[4;-7]

+disp(u,y)

+disp('The orthogonal projection of y on u=((y.u)/(u.u))*u')

+a=y'*u

+b=u'*u

+c=(a/b)*u

+disp(c,'=')

+disp('The component of y orthogonal to u is:')

+disp(y-c)
\ No newline at end of file
diff --git a/26/CH6/EX6.2.19/6_2_19.sce b/26/CH6/EX6.2.19/6_2_19.sce
new file mode 100755
index 000000000..b54516325
--- /dev/null
+++ b/26/CH6/EX6.2.19/6_2_19.sce
@@ -0,0 +1,9 @@
+disp('given vectors u and v are:')

+u=[-.6;.8]

+v=[.8;.6]

+disp(v,u)

+disp('u.v=')

+disp(u'*v)

+disp('Hence, {u v} is an orthogonal set.')

+disp('||u||=1 and ||v||=1')

+disp('Thus, {u v} is an orthonormal set')
\ No newline at end of file
diff --git a/26/CH6/EX6.2.7/6_2_7.sce b/26/CH6/EX6.2.7/6_2_7.sce
new file mode 100755
index 000000000..475ff9dc4
--- /dev/null
+++ b/26/CH6/EX6.2.7/6_2_7.sce
@@ -0,0 +1,20 @@
+disp('vectors u1 u2 and x are:')

+u1=[2;-3]

+u2=[6;4]

+x=[9;-7]

+disp(x,u2,u1)

+disp('u1.u2=')

+disp(u1'*u2)

+disp('u1.u2=0, {u1 u2} is an orthogonal set')

+disp('Hence {u1 u2} forms a basis of R2')

+disp('x can be written as: x=a*u1+b*u2')

+disp('where a=(x.u1)/(u1.u1)')

+a1=x'*u1

+a2=u1'*u1

+a=a1/a2

+disp(a,'=')

+disp('and b=(x.u2)/(u2.u2)')

+b1=x'*u2

+b2=u2'*u2

+b=b1/b2

+disp(b,'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.3.1/6_3_1.sce b/26/CH6/EX6.3.1/6_3_1.sce
new file mode 100755
index 000000000..424e4b954
--- /dev/null
+++ b/26/CH6/EX6.3.1/6_3_1.sce
@@ -0,0 +1,13 @@
+disp('Given vectors are:')

+u1=[0;1;-4;-1]

+u2=[3;5;1;1]

+u3=[1;0;1;-4]

+u4=[5;-3;-1;1]

+x=[10;-8;2;0]

+disp(x,'x=',u4,'u4=',u3,'u3=',u2,'u2=',u1,'u1=')

+disp('The vector in span{u4}=((x.u4)/(u4.u4))*u4')

+a1=x'*u4

+a2=u4'*u4

+disp((a1/a2)*u4)

+disp('Therefore, the vector in span{u1 u2 u3}=x-2*u4')

+disp(x-2*u4)
\ No newline at end of file
diff --git a/26/CH6/EX6.3.13/6_3_13.sce b/26/CH6/EX6.3.13/6_3_13.sce
new file mode 100755
index 000000000..3a3dcc9eb
--- /dev/null
+++ b/26/CH6/EX6.3.13/6_3_13.sce
@@ -0,0 +1,18 @@
+disp('Given vectors are:')

+v1=[2;-1;-3;1]

+v2=[1;1;0;-1]

+z=[3;-7;2;3]

+disp(z,'z=',v2,'v2=',v1,'v1=')

+a=v1'*v2

+disp(a,'v1.v2=')

+if(a==0)

+  disp('v1 and v2 are orthogonal')

+end

+disp('By best spproximation theorem, closest point in span{v1 v2} to z is the orthogonal projection')

+disp('=((z.v1)/(v1.v1))*v1+((z.v2)/(v2.v2))*v2')

+a1=z'*v1

+a2=v1'*v1

+b1=z'*v2

+b2=v2'*v2

+disp((a1/a2)*v1,'+',(b1/b2)*v2,'=')

+disp((a1/a2)*v1+(b1/b2)*v2,'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.3.19/6_3_19.sce b/26/CH6/EX6.3.19/6_3_19.sce
new file mode 100755
index 000000000..cb1d97266
--- /dev/null
+++ b/26/CH6/EX6.3.19/6_3_19.sce
@@ -0,0 +1,12 @@
+disp('By orthogonal decomposition theorem,')

+disp('u3 is the sum of a vector in W=span{u1 u2} and a vector v orthogonal to W')

+disp('To find v, given u1 and u2')

+u1=[1;1;-2]

+u2=[5;-1;2]

+disp(u2,'u2=',u1,'u1=')

+disp('Projection of u3 on W')

+disp('= (-1/3)*u1+(1/15)*u2')

+disp((-1/3)*u1+(1/15)*u2,'=')

+disp('v= u3-(projection of u3 on W)')

+disp((-1/3)*u1+(1/15)*u2,'-',[0;0;1],'=')

+disp([0;0;1]-((-1/3)*u1+(1/15)*u2),'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.3.7/6_3_7.sce b/26/CH6/EX6.3.7/6_3_7.sce
new file mode 100755
index 000000000..a0d8ffeeb
--- /dev/null
+++ b/26/CH6/EX6.3.7/6_3_7.sce
@@ -0,0 +1,17 @@
+disp('Vectors u1 u2 and y are')

+u1=[1;3;-2]

+u2=[5;1;4]

+y=[1;3;5]

+disp(y,'y=',u2,'u2=',u1,'u1=')

+disp('u1.u2=')

+a=u1'*u2

+disp(a,'=')

+disp('Hence, {u1 u2} form an orthogonal basis.')

+disp('Let W=span{u1 u2}')

+disp('Therefore, projection of y on W is:')

+disp('((y.u1)/(u1.u1))*u1+((y.u2)/(u2.u2))*u2')

+a1=y'*u1

+a2=u1'*u1

+b1=y'*u2

+b2=u2'*u2

+disp((b1/b2)*u2,'+',(a1/a2)*u1,'=')
\ No newline at end of file
diff --git a/26/CH6/EX6.4.1/6_4_1.sce b/26/CH6/EX6.4.1/6_4_1.sce
new file mode 100755
index 000000000..beeb5b0e0
--- /dev/null
+++ b/26/CH6/EX6.4.1/6_4_1.sce
@@ -0,0 +1,15 @@
+disp('to orthogonalise the given vectors using Gram-Schimdt orthogonalisation')

+x1=[3;0;-1]

+x2=[8;5;-6]

+disp(x2,'x2=',x1,'x1=')

+disp('Let v1=x1')

+v1=x1

+disp('v2=x2-((x2.v1)/(v1.v1))*v1')

+a1=x2'*v1

+a2=v1'*v1

+p=(a1/a2)*v1

+v2=x2-p

+disp(p,'-',x2,'=')

+disp(v2,'=')

+disp('Thus, an orthogonal basis is:')

+disp(v2,v1)
\ No newline at end of file
diff --git a/26/CH6/EX6.4.13/6_4_13.sce b/26/CH6/EX6.4.13/6_4_13.sce
new file mode 100755
index 000000000..a9834939f
--- /dev/null
+++ b/26/CH6/EX6.4.13/6_4_13.sce
@@ -0,0 +1,10 @@
+disp('QR decomposition of a matrix')

+disp('given matrix A=')

+a=[5 9;1 7;-3 -5;1 5]

+disp(a)

+disp('given matrix Q=')

+q=(1/6)*[5 -1;1 5;-3 1;1 3]

+disp(q)

+disp('Therefore, R=')

+s=q'*a

+disp(s)
\ No newline at end of file
diff --git a/26/CH6/EX6.4.7/6_4_7.sce b/26/CH6/EX6.4.7/6_4_7.sce
new file mode 100755
index 000000000..90d78f57a
--- /dev/null
+++ b/26/CH6/EX6.4.7/6_4_7.sce
@@ -0,0 +1,19 @@
+disp('to orthogonalise the given vectors using Gram-Schimdt orthogonalisation')

+x1=[2;-5;1]

+x2=[4;-1;2]

+disp(x2,'x2=',x1,'x1=')

+disp('Let v1=x1')

+v1=x1

+disp('v2=x2-((x2.v1)/(v1.v1))*v1')

+a1=x2'*v1

+a2=v1'*v1

+p=(a1/a2)*v1

+v2=x2-p

+disp(p,'-',x2,'=')

+disp(v2,'=')

+disp('Thus, an orthogonal basis is:')

+disp(v2,v1)

+disp('Normalizing v1 and v2, we get')

+s1=sqrt(v1(1,1)^2+v1(2,1)^2+v1(3,1)^2)

+s2=sqrt(v2(1,1)^2+v2(2,1)^2+v2(3,1)^2)

+disp(v2/s2,v1/s1)
\ No newline at end of file
diff --git a/26/CH6/EX6.5.1/6_5_1.sce b/26/CH6/EX6.5.1/6_5_1.sce
new file mode 100755
index 000000000..158a17efa
--- /dev/null
+++ b/26/CH6/EX6.5.1/6_5_1.sce
@@ -0,0 +1,15 @@
+disp('The co-efficient matrix is:')

+a=[-1 2;2 -3;-1 3]

+disp(a,'A=')

+disp('The RHS is:')

+b=[4;1;2]

+disp(b)

+disp('Product of transpose of A and A=')

+p1=a'*a

+disp(p1)

+disp('Product of transpose of A and b=')

+p2=a'*b

+disp(p2)

+disp('Hence, the solution is:')

+p=inv(p1)*p2

+disp(p)
\ No newline at end of file
diff --git a/26/CH6/EX6.5.13/6_5_13.sce b/26/CH6/EX6.5.13/6_5_13.sce
new file mode 100755
index 000000000..208e72660
--- /dev/null
+++ b/26/CH6/EX6.5.13/6_5_13.sce
@@ -0,0 +1,22 @@
+disp('To determine if u is the least square solution to Ax=b')

+disp('Given')

+a=[3 4;-2 1;3 4]

+disp(a,'A=')

+b=[11;-9;5]

+disp(b,'b=')

+u=[5;-1]

+v=[5;-2]

+disp(v,'v=',u,'u=')

+disp('Au=')

+disp(a*u)

+c=b-a*u

+disp(c,'b-Au=')

+disp('||b-Au||=')

+disp(sqrt(c(1,1)^2+c(2,1)^2+c(3,1)^2))

+disp('Av=')

+disp(a*v)

+d=b-a*v

+disp(d,'b-Av=')

+disp('||b-Av||=')

+disp(sqrt(d(1,1)^2+d(2,1)^2+d(3,1)^2))

+disp('Since Av is more closer to A than Au, u is not the least square solution.')
\ No newline at end of file
diff --git a/26/CH6/EX6.5.7/6_5_7.sce b/26/CH6/EX6.5.7/6_5_7.sce
new file mode 100755
index 000000000..aa8773a5f
--- /dev/null
+++ b/26/CH6/EX6.5.7/6_5_7.sce
@@ -0,0 +1,38 @@
+disp('The co-efficient matrix is:')

+a=[1 -2;-1 2;0 3;2 5]

+disp(a,'A=')

+disp('The RHS is:')

+b=[3;1;-4;2]

+disp(b,'b=')

+disp('Product of transpose of A and A=')

+p1=a'*a

+disp(p1)

+disp('Product of transpose of A and b=')

+p2=a'*b

+disp('Forming an augmented matrix to solve the normal equations')

+p=[p1 p2]

+disp(p)

+disp('performing row operations')

+disp('R2=R2-R1')

+p(2,:)=p(2,:)-p(1,:)

+disp(p)

+disp('R1=R1/6 and R2=R2/36')

+p(1,:)=p(1,:)/6

+p(2,:)=p(2,:)/36

+disp(p)

+disp('R1=R1-R2')

+p(1,:)=p(1,:)-p(2,:)

+disp(p)

+disp('Hence, the solution is:')

+disp(p(:,3))

+x=p(:,3)

+disp('The least square error is=||Ax-b||')

+disp('Ax-b=')

+disp(a*x-b)

+c=a*x-b

+s=0

+for i=1:4

+  s=s+c(i,1)^2

+end

+disp('||Ax-b||=')

+disp(sqrt(s))
\ No newline at end of file
diff --git a/26/CH6/EX6.6.1/6_6_1.sce b/26/CH6/EX6.6.1/6_6_1.sce
new file mode 100755
index 000000000..d61d53884
--- /dev/null
+++ b/26/CH6/EX6.6.1/6_6_1.sce
@@ -0,0 +1,18 @@
+disp('To obtain a least sqaure line from the given data')

+disp('Placing the x coordinates of the data in second column of matrix X we get:')

+x=[1 0;1 1;1 2;1 3]

+disp(x,'X=')

+disp('Placing the y coordinates in y vector')

+y=[1;1;2;2]

+disp(y,'y=')

+disp('Product of transpose of X and X=')

+p1=x'*x

+disp(p1)

+disp('Product of transpose of X and y=')

+p2=x'*y

+disp(p2)

+disp('The least square solution =')

+disp(inv(p1)*p2)

+p=inv(p1)*p2

+disp('Hence, the least square line is:')

+disp('x',p(2,1),'+',p(1,1),'=','y')
\ No newline at end of file
diff --git a/26/CH7/EX7.1.1/7_1_1.sce b/26/CH7/EX7.1.1/7_1_1.sce
new file mode 100755
index 000000000..09fe0b631
--- /dev/null
+++ b/26/CH7/EX7.1.1/7_1_1.sce
@@ -0,0 +1,8 @@
+disp('To check if the given 2X2 matrix is symmetric')

+a=[3 5;5 -7]

+disp(a,'A=')

+if(a(1,2)==a(2,1))

+  disp('A is a symmetric matrix because the (1,2) and(2,1) entries match.')

+else

+  disp('A is not a symmetric matrix')

+end
\ No newline at end of file
diff --git a/26/CH7/EX7.1.13/7_1_13.sce b/26/CH7/EX7.1.13/7_1_13.sce
new file mode 100755
index 000000000..25ee58c2c
--- /dev/null
+++ b/26/CH7/EX7.1.13/7_1_13.sce
@@ -0,0 +1,45 @@
+disp('To diagonalize the given matrix A')

+a=[3 1;1 3]

+disp(a,'A=')

+eig=spec(a)

+disp('Eigen values of A are:')

+disp(eig)

+disp('for lambda=4')

+disp('A-4I=')

+disp(a-4*eye(2,2))

+b=a-4*eye(2,2)

+disp('To find the eigenvector, form an augmented matrix.')

+c=[b [0;0]]

+disp('performing row operations')

+disp(c)

+c(2,:)=c(2,:)+c(1,:)

+disp(c)

+disp('With x2 as free variable, x1=x2')

+disp('Hence a basis for the eigenspace is:')

+d=[1;1]

+disp(d)

+disp('Upon normalizing')

+disp(d/(sqrt(2)))

+u1=d/(sqrt(2))

+disp('for lambda=2')

+disp('A-2I=')

+b=a-2*eye(2,2)

+disp(b)

+disp('To find the eigenvector, form an augmented matrix.')

+c=[b [0;0]]

+disp('performing row operations')

+disp(c)

+c(2,:)=c(2,:)-c(1,:)

+disp(c)

+disp('With x2 as free variable, x1=-x2')

+disp('Hence a basis for the eigenspace is:')

+d=[-1;1]

+disp(d)

+disp('Upon normalizing')

+disp(d/(sqrt(2)))

+u2=d/(sqrt(2))

+disp('Matrix P=')

+p=[u1 u2]

+disp(p)

+disp('The corresponding matrix D=')

+disp([eig(2,1) 0;0 eig(1,1)])
\ No newline at end of file
diff --git a/26/CH7/EX7.1.19/7_1_19.sce b/26/CH7/EX7.1.19/7_1_19.sce
new file mode 100755
index 000000000..5adfd2b7d
--- /dev/null
+++ b/26/CH7/EX7.1.19/7_1_19.sce
@@ -0,0 +1,64 @@
+disp('PD decomposition of a matrix A')

+a=[3 -2 4;-2 6 2;4 2 3]

+disp(a,'A=')

+disp('Eigenvalues of A are')

+eig=spec(a)

+disp(eig)

+disp(eig(2,1),'for lambda =')

+disp('A-(lambda)I=')

+b=a-eig(2,1)*eye(3,3)

+disp(b)

+disp('To find eigenvector, form an augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+disp('performing row operations')

+c(2,:)=c(2,:)-(c(2,1)/c(1,1))*c(1,:)

+c(3,:)=c(3,:)-(c(3,1)/c(1,1))*c(1,:)

+disp(c)

+disp('With x2 and x3 as free variables, we get two vectors.')

+disp('x1=-.5x2+x3')

+disp('Thus, the two vectors are')

+v1=[-1;2;0]

+v2=[1;0;1]

+disp(v2,v1)

+disp('Orthogonalizing v1 and v2')

+disp('Let x1=v1')

+disp('x2=v2-((v2.v1)/(v1.v1))*v1')

+x1=v1

+a1=v2'*v1

+a2=v1'*v1

+x2=v2-(a1/a2)*v1

+x1=x1/(sqrt(x1(1,1)^2+x1(2,1)^2+x1(3,1)^2))

+x1=x2/(sqrt(x2(1,1)^2+x2(2,1)^2+x2(3,1)^2))

+disp('An orthonormal basis is:')

+disp(x2,x1)

+disp(eig(1,1),'for lambda=')

+disp('A-(lambda)I=')

+b=a-eig(1,1)*eye(3,3)

+disp(b)

+disp('To find eigenvector, form an augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+disp('performing row operations')

+c(2,:)=c(2,:)-(c(2,1)/c(1,1))*c(1,:)

+c(3,:)=c(3,:)-(c(3,1)/c(1,1))*c(1,:)

+disp(c)

+c(3,:)=c(3,:)-(c(3,2)/c(2,2))*c(2,:)

+disp(c)

+c(1,:)=c(1,:)/c(1,1)

+c(2,:)=c(2,:)/c(2,2)

+disp(c)

+c(1,:)=c(1,:)-(c(1,2)/c(2,2))*c(2,:)

+disp(c)

+disp('With x3 as free variable')

+disp('x1=x3 and x2=-.5x3')

+disp('Thus a basis for the eigenspace is:')

+v3=[1;-.5;1]

+disp(v3)

+disp('upon normalizing')

+v3=v3/(sqrt(v3(1,1)^2+v3(2,1)^2+v3(3,1)^2))

+disp(v3)

+disp('Thus, matrix P=')

+disp([x1 x2 v3])

+disp('Corresponding matrix D=')

+disp([eig(2,1) 0 0;0 eig(3,1) 0;0 0 eig(1,1)])
\ No newline at end of file
diff --git a/26/CH7/EX7.1.7/7_1_7.sce b/26/CH7/EX7.1.7/7_1_7.sce
new file mode 100755
index 000000000..8a19473d1
--- /dev/null
+++ b/26/CH7/EX7.1.7/7_1_7.sce
@@ -0,0 +1,21 @@
+disp('To show that the given matrix P is orthogonal.')

+p=[.6 .8;.8 -.6]

+disp(p,'P=')

+disp('P is composed of two vectors.')

+p1=[.6;.8]

+p2=[.8;-.6]

+disp(p2,'p2=',p1,'p1=')

+disp('To show that the columns are orthonormal')

+disp('p1.p2=')

+s=p1'*p2

+r=p1'

+disp(p2,'*',r,'=')

+disp(s,'=')

+if(s==0)

+  disp('The columns of P are othonormal')

+end

+disp('||p1||=')

+disp(sqrt(p(1,1)^2+p(2,1)^2))

+disp('||p2||=')

+disp(sqrt(p(1,2)^2+p(2,2)^2))

+disp('Hence, ||p1||=||p2||=1. Thus P is an orthogonal matrix')
\ No newline at end of file
diff --git a/26/CH7/EX7.2.1/7_2_1.sce b/26/CH7/EX7.2.1/7_2_1.sce
new file mode 100755
index 000000000..5be83ee16
--- /dev/null
+++ b/26/CH7/EX7.2.1/7_2_1.sce
@@ -0,0 +1,14 @@
+disp('given matrix A and vector x')

+a=[5 (1/3);(1/3) 1]

+disp(a,'A=')

+x=[6;1]

+disp(x,'x=')

+disp('Product of transpose of x and A and x=')

+p=x'*a*x

+disp(p)

+disp('New value of vector x=')

+x=[1;3]

+disp(x)

+disp('Product of transpose of x and A and x=')

+p=x'*a*x

+disp(p)
\ No newline at end of file