Merge pull request #1 from prashantsinalkar/masterHEADmaster

Added R TBC

23 files changed, 464 insertions, 0 deletions

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.1/Ex6.1_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.1/Ex6.1_1.r
new file mode 100644
index 00000000..bb43e3c8
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.1/Ex6.1_1.r
@@ -0,0 +1,19 @@
+# Example : 1   Chapter : 6.1     Page No: 284

+# Eigen values and eigen vectors

+A<-matrix(c(0.8,0.2,0.3,0.7),ncol=2)

+sol<-eigen(A)

+lambda<-sol$values

+x<-sol$vectors

+print("The eigen values of the matrix are")

+print(lambda)

+print("The eigen vectors of the matrix in normalised form are")

+print(x)

+#to get eigen vectors in the textbook multiply normalised vectors by scalars

+x[,1]<-x[,1]*(0.6/x[1,1])

+x[,2]<-x[,2]*(1/x[1,2])

+print("Eigen vectors with respect to the above eigen values respectively are")

+print(x)

+print(A%*%x)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.2/Ex6.1_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.2/Ex6.1_2.r
new file mode 100644
index 00000000..369a5e47
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.2/Ex6.1_2.r
@@ -0,0 +1,13 @@
+# Example : 2   Chapter : 6.1     Page No: 285

+# Eigen values and eigen vectors of Projection matrix

+A<-matrix(c(0.5,0.5,0.5,0.5),ncol=2)

+sol<-eigen(A)

+lambda<-sol$values

+x<-sol$vectors #These are normalised eigen vectors 

+#to get eigen vectors in text book multiply them with scalars

+x[,1]<-x[,1]*(1/x[1,1])

+x[,2]<-x[,2]*(1/x[1,2])

+print("The eigen values of the matrix are")

+print(lambda)

+print("The eigen vectors of the matrix are")

+print(x)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.3/Ex6.1_3.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.3/Ex6.1_3.r
new file mode 100644
index 00000000..f878ef16
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.3/Ex6.1_3.r
@@ -0,0 +1,8 @@
+# Example : 3   Chapter : 6.1     Page No: 286

+# Eigen values and eigen vectors of Reflection matrix

+A<-matrix(c(0,1,1,0),ncol=2)

+sol<-eigen(A)

+lambda<-sol$values

+print("The eigen values of the matrix are")

+print(lambda)

+#The  answer may slightly vary due to  rounding off values

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.4/Ex6.1_4.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.4/Ex6.1_4.r
new file mode 100644
index 00000000..6d8b139e
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.4/Ex6.1_4.r
@@ -0,0 +1,81 @@
+# Packages used  : pracma

+# To install pracma,type following in command line while connected to internet

+# install.packages("pracma") 

+# package can be included by command " library(pracma) "

+# for more information about pracma visit https://cran.r-project.org/web/packages/pracma/index.html

+

+# Example : 4   Chapter : 6.1     Page No: 287

+# Eigen values and eigen vectors of Singualar matrix

+

+library(pracma)

+nullspacebasis <- function(A){

+  R<-rref(A)

+  m<-nrow(A)

+  n<-ncol(A)

+  pivotcol<-c() #vector to store the column numbers of pivot columns

+  freecol<-c()  #vector to store the column numbers of free columns

+  i<-1

+  j<-1

+  

+  # to find which columns are pivot and which are free

+  while(i<=m & j<=n){

+    if(R[i,j]==1){

+      pivotcol<-c(pivotcol,j)

+      i<-i+1

+      j<-j+1

+    }

+    else{

+      j<-j+1

+    }

+  }

+  y<-length(pivotcol)

+  freecol<-c(1:n)

+  freecol<-freecol[!freecol%in%pivotcol]

+  x<-length(freecol)

+  N<-c()

+  #find the basis for null space based on Row reduced echelon form of given matrix

+  if(y==n){

+    return(N)

+  }

+  for(i in 1:x){

+    temp<-c(1:n)

+    for(j in 1:x){

+      temp[freecol[j]]<-0

+    }

+    temp[freecol[i]]<-1

+    temp[freecol[i]]

+    for(j in 1:y){

+      temp[pivotcol[j]]<-R[j,freecol[i]]*-1

+    }

+    N<-c(N,temp)

+  }

+  N<-matrix(N,nrow=n,ncol=x)

+  #Basis for the nullspace of given matrix

+  return(N)

+}

+A<-matrix(c(1,2,2,4),ncol=2)

+sol<-eigen(A)

+print(sol)

+lambda<-sol$values

+print("The eigen values of the matrix are")

+print(lambda)

+I<-matrix(c(1,0,0,1),ncol=2)

+E1<-A-lambda[1]*I

+E1

+rref(E1)

+x1<-nullspacebasis(E1)

+E2<-A-lambda[2]*I

+rref(E2)

+x2<-nullspacebasis(E2)

+print("The eigen vectors of the matrix in normalized form are")

+print(x1)

+print(x2)

+#to get eigen vectors in the textbook multiply normalised vectors by scalars

+x1<-2*x1

+x2<--1*x2

+print("The eigen vectors of the matrix are ")

+print(x2)

+print(x1)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.a/Ex6_6.1A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.a/Ex6_6.1A.r
new file mode 100644
index 00000000..b7bf991a
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.a/Ex6_6.1A.r
@@ -0,0 +1,31 @@
+# Example : 6.1A   Chapter : 6.1     Page No: 291

+# Eigen values and eigen vectors

+solution<-function(A){

+  sol<-eigen(A)

+  lambda<-sol$values

+  x<-sol$vectors #these are normalised eigen vectors

+  #to get the eigen vectors as in texxt book multiply these normalised vectors with scalars

+  x[,1]<-x[,1]*(1/x[1,1])

+  x[,2]<-x[,2]*(1/x[1,2])

+  print("The eigen values of the matrix are")

+  print(lambda)

+  print("The eigen vecotrs of the matrix respective to above eigen values are")

+  print(x)

+}

+

+A<-matrix(c(2,-1,-1,2),ncol=2)

+print("For A")

+solution(A)

+A2<-A%*%A

+print("For square of A")

+solution(A2)

+A1<-solve(A)

+print("For inverse of A")

+solution(A1)

+I<-matrix(c(1,0,0,1),ncol=2)

+A4I<-A+4*I

+print("For A+4I")

+solution(A4I)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.b/Ex6_6.1B.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.b/Ex6_6.1B.r
new file mode 100644
index 00000000..f2c11521
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.1.b/Ex6_6.1B.r
@@ -0,0 +1,19 @@
+# Example : 6.1B   Chapter : 6.1     Page No: 292

+# Eigen values and eigen vectors

+solution<-function(A){

+  sol<-eigen(A)

+  lambda<-round(sol$values)

+  x<-sol$vectors #these are normalised eigen vectors

+  #to get eigen vectors in text book multiply normalised eigen vectors with scalars

+  x[,1]<-x[,1]*(1/x[1,1])

+  x[,2]<-round(x[,2]*(1/x[1,2]))

+  x[,3]<-x[,3]*(1/x[1,3])

+  print("The eigen values of the matrix are")

+  print(lambda)

+  print("The eigen vectors of the matrix respective to above eigen values are")

+  print(x)

+}

+

+A<-matrix(c(1,-1,0,-1,2,-1,0,-1,1),ncol=3)

+solution(A)

+#The  answer may slightly vary due to  rounding off values

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.1/Ex6.2_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.1/Ex6.2_1.r
new file mode 100644
index 00000000..42b08fae
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.1/Ex6.2_1.r
@@ -0,0 +1,20 @@
+# Example : 1   Chapter : 6.2     Page No: 299

+# Diagonilizing a Matrix and Power of matrix computed from power of its diagonal matrix

+A<-matrix(c(1,0,5,6),ncol=2)

+lambda<-eigen(A)$values

+S<-eigen(A)$vectors

+S<-round(S)

+S1<-solve(S)

+Diag_matrix<-round(S1%*%A%*%S)

+print("Diagonal Matrix is ")

+print(Diag_matrix)

+A2<-A%*%A

+print("The square of matrix")

+print(A2)

+print("The Power of matrix can also be computed from its diagonal matrix as follows")

+A2_diag<-S%*%Diag_matrix%*%Diag_matrix%*%S1

+print("The square of matrix computed from its diagonal matrix")

+print(A2_diag)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.2/Ex6.2_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.2/Ex6.2_2.r
new file mode 100644
index 00000000..09d7f4c4
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.2/Ex6.2_2.r
@@ -0,0 +1,16 @@
+# Example : 2   Chapter : 6.2     Page No: 300

+# The diagonal matrix of any matrix contains the eigen values in its main diagonal

+A<-matrix(c(0.8,0.2,0.3,0.7),ncol=2)

+lambda<-eigen(A)$values

+print(lambda)

+S<-eigen(A)$vectors #Normlised eigen vectors, Answer can also be validated with normalised eigen vectors

+#to get eigen vector matrix in text book

+S[,1]<-S[,1]*(0.6/S[1,1])

+S[,2]<-S[,2]*(1/S[1,2])

+S1<-solve(S)

+Diag_matrix<-diag(2)*lambda

+print("S*diag_matrix(A)*S-1 is A")

+print(S%*%Diag_matrix%*%S1)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.b/Ex6_6.2B.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.b/Ex6_6.2B.r
new file mode 100644
index 00000000..991d615a
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.2.b/Ex6_6.2B.r
@@ -0,0 +1,33 @@
+# Packages used  : pracma

+# To install pracma,type following in command line while connected to internet

+# install.packages("pracma") 

+# package can be included by command " library(pracma) "

+# for more information about pracma visit https://cran.r-project.org/web/packages/pracma/index.html

+

+# Example : 6.2B   Chapter : 6.2     Page No: 306

+# Inverse Eigen values Determinant and eigen vector matrix

+library(pracma)

+A<-5*eye(4)-ones(4)

+eigenvalues<-eigen(A)$values

+print("Eigen Values of A ")

+print(eigenvalues)

+A1<-solve(A)

+print("Inverse of A ")

+print(A1)

+eigenvalues1<-eigen(A1)$value

+print("Eigen Values of A inverse ")

+print(eigenvalues1)

+detA<-det(A)

+print("Determinant of A")

+print(detA)

+x<-eigen(A)$vectors[,4] #normalized  eigen vector of eigen value=1

+#toget eigen vector in text book

+x<-x*(1/x[1])

+print("Eigen vector of A for eigen value 1")

+print(x)

+print("The other eigen vectors are perpendicular to x since A is Symmetric so eigen vector matrix contains x with different signs as follows")

+S<-matrix(c(1,-1,1,-1,1,1,-1,-1,1,-1,-1,1,x),ncol=4)

+print(S)

+print("To get normalized matrix multiply by magnitude of vectors which is same for all and is 0.5")

+S<-0.5*S

+print(S) # The eigen vectors are with respect to 5,5,5,1
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.1/Ex6.3_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.1/Ex6.3_1.r
new file mode 100644
index 00000000..72bb740f
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.1/Ex6.3_1.r
@@ -0,0 +1,22 @@
+# Example : 1   Chapter : 6.3     Page No: 313

+# Solve Differential equation

+

+# lambda1,lamda2,x1,x2,c,d are computed here.. for remaining details look textbook

+A<-matrix(c(0,1,1,0),ncol=2)

+lambda<-eigen(A)$values

+x<-eigen(A)$vectors #These are normalised eigen vectors

+#to get eigen vectors in textbook .. Multiply them with the scalars

+x[,1]<-x[,1]*(1/x[1,1])

+x[,2]<-x[,2]*(1/x[1,2])

+print(x)

+u<-c(4,2)

+cd<-solve(x,u)

+C<-cd[1]

+D<-cd[2]

+print("Lambda 1 and Lambda 2")

+print(lambda)

+print("x1 and x2")

+print(x)

+print("C and D are")

+print(C)

+print(D)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.2/Ex6.3_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.2/Ex6.3_2.r
new file mode 100644
index 00000000..d583fb2f
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.2/Ex6.3_2.r
@@ -0,0 +1,18 @@
+# Example : 2   Chapter : 6.3     Page No: 314

+# Solve Differential equation

+

+# lambda1,lamda2,lambda3,x1,x2,x3,c1,c2,c3 are computed here.. for remaining details look textbook

+A<-matrix(c(1,0,0,1,2,0,1,1,3),ncol=3)

+lambda<-eigen(A)$values

+x<-round(eigen(A)$vectors)

+u<-c(9,7,4)

+c<-solve(x,u)

+print("Lambda 3 and Lambda 2 and Lambda 1 are")

+print(lambda)

+print("x3 and x2 and x1")

+print(x)

+print("c3,c2 and c1 are")

+print(c)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.6/Ex6.3_6.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.6/Ex6.3_6.r
new file mode 100644
index 00000000..dad9885d
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.6/Ex6.3_6.r
@@ -0,0 +1,18 @@
+# Example : 6   Chapter : 6.3     Page No: 321
+# Solve Differential equation
+
+# lambda1,lamda2,x1,x2,c,d are computed here.. for remaining details look textbook
+A<-matrix(c(1,0,1,2),ncol=2)
+lambda<-eigen(A)$values
+x<-round(eigen(A)$vectors)
+u<-c(2,1)
+c<-solve(x,u)
+print("Lambda 1 and Lambda 2")
+print(lambda)
+print("x1 and x2")
+print(x)
+print("c1 and c2 are")
+print(c)
+#The  answer may slightly vary due to  rounding off values
+#The answers provided in the text book may vary because of the computation process
+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.b/Ex6_6.3B.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.b/Ex6_6.3B.r
new file mode 100644
index 00000000..30f54de1
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.3.b/Ex6_6.3B.r
@@ -0,0 +1,18 @@
+# Example : 6.3B   Chapter : 6.3     Page No: 322

+# Eigen Values and Eigen vectors of A

+A<-matrix(c(-2,1,0,1,-2,1,0,1,-2),ncol=3)

+eigenvalues<-eigen(A)$values

+x<-eigen(A)$vectors

+print("Eigen values of A are ")

+print(eigenvalues)

+print("Eigen vectors of A in normalised form")

+print(x)

+#to get eigen vectors in the textbook multiply normalised vectors by scalars

+x[,1]<-x[,1]*(1/x[1,1])

+x[,2]<-x[,2]*(1/x[1,2])

+x[,3]<-x[,3]*(1/x[1,3])

+print("Eigen vectors with respect to the above eigen values respectively are")

+print(x)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.1/Ex6.4_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.1/Ex6.4_1.r
new file mode 100644
index 00000000..9baefa7d
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.1/Ex6.4_1.r
@@ -0,0 +1,14 @@
+# Example : 1   Chapter : 6.4     Page No: 331

+# Eigen Values and Eigen vectors of A

+A<-matrix(c(1,2,2,4),ncol=2)

+lambda<-eigen(A)$values

+x<-eigen(A)$vectors #These are Normalised eigen vectors

+#to get eigen vectors in textbook

+x[,1]<-x[,1]*(1/x[1,1])

+x[,2]<-x[,2]*(2/x[1,2])

+print("Eigen values and eigen vectors of respective eigen values of A")

+print(lambda)

+print(x)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.4/Ex6.4_4.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.4/Ex6.4_4.r
new file mode 100644
index 00000000..423a09e1
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.4/Ex6.4_4.r
@@ -0,0 +1,23 @@
+# Packages used  : pracma

+# To install pracma,type following in command line while connected to internet

+# install.packages("pracma") 

+# package can be included by command " library(pracma) "

+# for more information about pracma visit https://cran.r-project.org/web/packages/pracma/index.html

+

+# Example : 4   Chapter : 6.4     Page No: 334

+# Pivots and Eigen values have same signs for the Symmetric matrices

+library(pracma)

+A<-matrix(c(1,3,3,1),ncol=2)

+u<-lu(A)$U

+pivots<-c(u[1,1],u[2,2])

+eigenvalues<-eigen(A)$values

+print("Eigen values and pivots have same signs for symmetric matrix")

+print(pivots)

+print(eigenvalues)

+print("This is not true if the matrix is not symmetric")

+B<-matrix(c(1,-1,6,-4),ncol=2)

+Bu<-lu(B)$U

+Bpivots<-c(Bu[1,1],Bu[2,2])

+Beigenvalues<-eigen(B)$values

+print(Bpivots)

+print(Beigenvalues)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.b/Ex6_6.4B.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.b/Ex6_6.4B.r
new file mode 100644
index 00000000..566bf027
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.4.b/Ex6_6.4B.r
@@ -0,0 +1,32 @@
+# Example : 6.4B   Chapter : 6.4     Page No: 336

+# Eigen Values and Eigen vectors of given matrices

+A3<-matrix(c(2,-1,0,-1,2,-1,0,-1,2),ncol=3)

+eigenvalues<-eigen(A3)$values

+x<-eigen(A3)$vectors

+print("Eigen values of A3 are ")

+print(eigenvalues)

+print("Eigen vectors of A3 in normalised form are ")

+print(x)

+#to get eigen vectors in the textbook multiply normalised vectors by scalars

+x[,1]<-x[,1]*(1/x[1,1])

+x[,2]<-x[,2]*(sqrt(2)/x[1,2])

+x[,3]<-x[,3]*(1/x[1,3])

+print("Eigen vectors with respect to the above eigen values respectively are")

+print(x)

+B4<-matrix(c(1,-1,0,0,-1,2,-1,0,0,-1,2,-1,0,0,-1,1),ncol=4)

+B4eigenvalues<-eigen(B4)$values

+B4x<-eigen(B4)$vectors

+print("Eigen values of B4 are ")

+print(B4eigenvalues)

+print("Eigen vectors of B4 in normalised form are")

+print(B4x)

+#to get eigen vectors in the textbook multiply normalised vectors by scalars

+B4x[,1]<-B4x[,1]*(1/B4x[1,1])

+B4x[,2]<-B4x[,2]*(1/B4x[1,2])

+B4x[,3]<-B4x[,3]*(1/B4x[1,3])

+B4x[,4]<-B4x[,4]*(1/B4x[1,4])

+print("Eigen vectors with respect to the above eigen values respectively are")

+print(B4x)

+#The  answer may slightly vary due to  rounding off values

+#The answers provided in the text book may vary because of the computation process

+#Both answers are correct , here it is taken -Ax+b=0 , In the text book it is considered as Ax-b=0

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.5.1/Ex6.5_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.5.1/Ex6.5_1.r
new file mode 100644
index 00000000..e9a01511
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.5.1/Ex6.5_1.r
@@ -0,0 +1,18 @@
+# Packages used  : pracma

+# To install pracma,type following in command line while connected to internet

+# install.packages("pracma") 

+# package can be included by command " library(pracma) "

+# for more information about pracma visit https://cran.r-project.org/web/packages/pracma/index.html

+

+# Example : 1   Chapter : 6.5     Page No: 344

+# Tests for positive definitenes

+library(pracma)

+A<-matrix(c(2,-1,0,-1,2,-1,0,-1,2),ncol=3)

+u<-lu(A)$U

+pivots<-c(u[1,1],u[2,2],u[3,3])

+upper_left_deteminants<-c(A[1,1],det(A[-3,-3]),det(A))

+eigenvalues<-eigen(A)$values

+print("All pivots, upper left determinants and eigen values are positive")

+print(pivots)

+print(upper_left_deteminants)

+print(eigenvalues)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.1/Ex6.6_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.1/Ex6.6_1.r
new file mode 100644
index 00000000..f385fe21
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.1/Ex6.6_1.r
@@ -0,0 +1,17 @@
+# Example : 1   Chapter : 6.6     Page No: 356
+# Similar matrices are matrices with same eigen values
+A<-matrix(c(0.5,0.5,0.5,0.5),ncol=2)
+Aev<-eigen(A)$values
+print("Eigen values of A ")
+print(Aev)
+Lambda<-matrix(c(1,0,0,0),ncol=2)
+Lev<-eigen(Lambda)$values
+print("eigen values of lambda matrix")
+print(Lev)
+M<-matrix(c(1,1,0,2),ncol=2)
+M1<-solve(M)
+M1AM<-M1%*%A%*%M
+M1AMev<-eigen(M1AM)$values
+print("EIgen values of M-1*A*M")
+print(M1AMev)
+print("A and M-1*A*M are similar matrices")
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.2/Ex6.6_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.2/Ex6.6_2.r
new file mode 100644
index 00000000..91ec2c0d
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.2/Ex6.6_2.r
@@ -0,0 +1,10 @@
+# Example : 2   Chapter : 6.6     Page No: 356
+# Similar matrices  with repeated eigen values
+A<-matrix(c(0,0,1,0),ncol=2)
+Aev<-eigen(A)$values
+A1<-matrix(c(1,1,-1,-1),ncol=2)
+A1ev<-round(eigen(A1)$values)
+print("Both th egiven matrices are similar because their eigen values are same")
+print(Aev)
+print(A1ev)
+#The  answer may slightly vary due to  rounding off values
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.3/Ex6.6_3.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.3/Ex6.6_3.r
new file mode 100644
index 00000000..fb2496ae
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.3/Ex6.6_3.r
@@ -0,0 +1,9 @@
+# Example : 3   Chapter : 6.6     Page No: 357
+# Jordans theorem and Jordan Matrix
+J<-matrix(c(5,0,0,1,5,0,0,1,5),ncol=3)
+Jev<-eigen(J)$values
+JT<-t(J)
+JTev<-eigen(JT)$values
+print("Jordans theorem says that both J and transpose of J are similar")
+print(Jev)
+print(JTev)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.a/Ex6_6.6A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.a/Ex6_6.6A.r
new file mode 100644
index 00000000..58eccd09
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.6.a/Ex6_6.6A.r
@@ -0,0 +1,9 @@
+# Example : 3   Chapter : 6.6     Page No: 357
+# Pascals matrix and its inverse are similar
+A<-matrix(c(1,1,1,1,0,1,2,3,0,0,1,3,0,0,0,1),ncol=4)
+A1<-solve(A)
+Aev<-eigen(A)$values
+A1ev<-eigen(A1)$values
+print("Both pascal matrix and its inverse have same eigen values")
+print(Aev)
+print(A1ev)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.3/Ex6.7_3.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.3/Ex6.7_3.r
new file mode 100644
index 00000000..b61c47aa
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.3/Ex6.7_3.r
@@ -0,0 +1,8 @@
+# Example : 3   Chapter : 6.7     Page No: 366
+# Singular Value decomposition
+
+A<-matrix(c(2,-1,2,1),ncol=2)
+print("Singular value decomposition is given by")
+print(svd(A))
+#The  answer may slightly vary due to  rounding off values
+#The answers provided in the text book may vary because of the computation method followed.
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.4/Ex6.7_4.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.4/Ex6.7_4.r
new file mode 100644
index 00000000..ecd3e15f
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH6/EX6.7.4/Ex6.7_4.r
@@ -0,0 +1,8 @@
+# Example : 3   Chapter : 6.7     Page No: 366
+# Singular Value decomposition
+
+A<-matrix(c(2,1,2,1),ncol=2)
+print("Singular value decomposition is given by")
+print(svd(A))
+#The  answer may slightly vary due to  rounding off values
+#The answers provided in the text book may vary because of the computation method followed.