Merge pull request #1 from prashantsinalkar/masterHEADmaster

Added R TBC

14 files changed, 738 insertions, 0 deletions

diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.2/Ex3.2_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.2/Ex3.2_2.r
new file mode 100644
index 00000000..fda61595
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.2/Ex3.2_2.r
@@ -0,0 +1,59 @@
+# 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 : 2   Chapter : 3.2     Page No: 132
+# Find the Nullspace of given singular 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(1,3,2,6),nrow=2,ncol=2)
+N<-nullspacebasis(A)
+print("Basis vectors for nullspace of given matrix is")
+N
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.3/Ex3.2_3.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.3/Ex3.2_3.r
new file mode 100644
index 00000000..49568620
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.3/Ex3.2_3.r
@@ -0,0 +1,69 @@
+# 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 : 3   Chapter : 3.2     Page No: 133
+# Find the Nullspace of given matrices A,B,C
+
+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(1,3,2,8),ncol=2,nrow=2)
+N<-nullspacebasis(A)
+print("Basis vectors of the nullspace of matrix A is")
+N
+B<-rbind(A,2*A)
+N1<-nullspacebasis(B)
+print("Basis vectors of the nullspace of matrix B is")
+N1
+C<-cbind(A,2*A)
+C
+N2<-nullspacebasis(C)
+print("Basis vectors of the nullspace of matrix C is")
+N2
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.4/Ex3.2_4.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.4/Ex3.2_4.r
new file mode 100644
index 00000000..ccf644a4
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.4/Ex3.2_4.r
@@ -0,0 +1,60 @@
+# 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 : 3.2     Page No: 136
+# Find the Nullspace of given Upper traingular 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)
+}
+
+U<-matrix(c(1,0,5,0,7,9),nrow=2)
+N<-nullspacebasis(U)
+print("Basis vectors for the Nullspace of given upper traingular matrix is ")
+N
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.a/Ex3_3.2A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.a/Ex3_3.2A.r
new file mode 100644
index 00000000..a181d2de
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.a/Ex3_3.2A.r
@@ -0,0 +1,14 @@
+# Example : 3.2A   Chapter : 3.2     Page No: 139

+# Find the Matrix having the given special solutions

+s1<-c(-3,1,0,0)

+s2<-c(-2,0,-6,1)

+R<-diag(4)

+R<-R[-4,] 

+#As 1st column is pivot column it is not needed to change

+#As 3rd column is next pivot column , row reduced echolon form has 1 in second row of third column

+R[,3]<-R[,2]

+#Two free columns are modified acording to special solutions

+R[,2]<-c(-1*s1[1],0,0)

+R[,4]<-c(-1*s2[1],-1*s2[3],0)

+print("matrix having given special solutions")

+print(R)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.b/Ex3_3.2b.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.b/Ex3_3.2b.r
new file mode 100644
index 00000000..f6b24099
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.2.b/Ex3_3.2b.r
@@ -0,0 +1,73 @@
+# 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 : 3.2B   Chapter : 3.2     Page No: 140
+# Find the Specialsolution, Pivotcolumns, freecolumns and Reduced row echelon form for each given matrix
+
+library(pracma)
+solution <- 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){
+    N<-c()
+  }
+  else{
+  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
+  print("Special solutions are given by the basis vectors of null space")
+  print(N)
+  print("Pivot columns are")
+  print(pivotcol)
+  print("Free columns or free variables are")
+  print(freecol)
+  print("Row reduced echelon form is")
+  print(R)
+}
+A1<-matrix(c(0,0,0,0,0,0,0,0),nrow=2)
+print("For the matrix A1")
+solution(A1)
+A2<-matrix(c(3,1,6,2),nrow=2)
+print("For the matrix A2")
+solution(A2)
+A3<-cbind(A2,A2)
+print("For the matrix A3")
+solution(A3)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.a/Ex3_3.3A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.a/Ex3_3.3A.r
new file mode 100644
index 00000000..5dbfb740
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.a/Ex3_3.3A.r
@@ -0,0 +1,64 @@
+# 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 : 3.3A   Chapter : 3.3     Page No: 149
+# Find the row reduced echelon form, rank and special solution for given matrix
+
+library(pracma)
+solution <- 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){
+    N<-c()
+  }
+  else{
+  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)
+  }
+  print("Row reduced echelon form is")
+  print(R)
+  print("Rank of the Matrix is")
+  print(y)
+  print("Special solutions for given matrix are given by")
+  print(N)
+}
+
+A<-matrix(c(1,-1,0,0,-1,2,-1,0,0,-1,2,-1,0,0,-1,1),nrow=4)
+solution(A)
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.c/Ex3_3.3C.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.c/Ex3_3.3C.r
new file mode 100644
index 00000000..8b9d419d
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.3.c/Ex3_3.3C.r
@@ -0,0 +1,69 @@
+# 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 : 3.3C   Chapter : 3.3     Page No: 151
+# Find the special solutions and row reduced echelon forms
+
+library(pracma)
+solution <- 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){
+    N<-c()
+  }
+  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)
+  print("row reduced echelon form is")
+  print(R)
+  print("Special solution is ")
+  print(N)
+}
+#c is not equal to 4
+A<-matrix(c(1,3,4,2,6,8,1,3,2),ncol=3)
+solution(A)
+#c=4
+A4<-matrix(c(1,3,4,2,6,8,1,3,4),ncol=3,nrow=3)
+solution(A4)
+#c is not equal to 0
+B<-matrix(c(1,1,1,1),nrow=2)
+solution(B)
+#c = 0
+B0<-matrix(c(0,0,0,0),nrow=2)
+solution(B0)
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.2/Ex3.4_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.2/Ex3.4_2.r
new file mode 100644
index 00000000..e14008d0
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.2/Ex3.4_2.r
@@ -0,0 +1,67 @@
+# 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 : 2   Chapter : 3.4     Page No: 158
+# Find the particular solution and special solution of the given system
+
+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(1,1,1,2,1,-1),nrow=2)
+b<-c(3,4)
+s<-nullspacebasis(A)
+Ab<-cbind(A,b)
+Rd<-rref(Ab)
+p<-Rd[,4]
+p<-c(p,0)
+print("Particualar solution is")
+print(p)
+print("special solution is ")
+print(s)
+print("The complete solution = particular solution + special solution")
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.a/Ex3_3.4A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.a/Ex3_3.4A.r
new file mode 100644
index 00000000..f38ad725
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.a/Ex3_3.4A.r
@@ -0,0 +1,72 @@
+# 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 : 3.4A   Chapter : 3.4     Page No: 160
+# Find the Complete solution of the given system
+
+library(pracma)
+completesolution <- function(A,b){
+  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){
+    N<-c()
+  }
+  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)
+  s<-N
+  Ab<-cbind(A,b)
+  Rd<-rref(Ab)
+  temp<-Rd[,n+1]
+  p<-c(1:n)
+  for(i in 1:length(freecol)){
+    p[freecol[i]]<-0
+  }
+  for(i in 1:length(pivotcol)){
+    p[pivotcol[i]]<-temp[i]
+  }
+  print("The special solution is")
+  print(s)
+  print("The particular solution is")
+  print(p)
+  print("Complete solution = Particular solution + Special solution")
+}
+A<-matrix(c(1,2,3,2,4,6,3,8,7,5,12,13),nrow=3)
+b<-c(0,6,-6)
+completesolution(A,b)
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.c/Ex3_3.4C.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.c/Ex3_3.4C.r
new file mode 100644
index 00000000..1d77f3aa
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.4.c/Ex3_3.4C.r
@@ -0,0 +1,72 @@
+# 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 : 3.4C   Chapter : 3.4     Page No: 162
+# Find the Complete solution of the given system
+
+library(pracma)
+completesolution <- function(A,b){
+  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){
+    N<-c()
+  }
+  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)
+  s<-N
+  Ab<-cbind(A,b)
+  Rd<-rref(Ab)
+  temp<-Rd[,n+1]
+  p<-c(1:n)
+  for(i in 1:length(freecol)){
+    p[freecol[i]]<-0
+  }
+  for(i in 1:length(pivotcol)){
+    p[pivotcol[i]]<-temp[i]
+  }
+  print("The special solution is")
+  print(s)
+  print("The particular solution is")
+  print(p)
+  print("Complete solution = particular solution + Special solution")
+}
+A<-matrix(c(1,2,4,2,4,8,1,4,6,0,8,8),nrow=3)
+b<-c(4,2,10)
+completesolution(A,b)
\ No newline at end of file
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.5.1/Ex3.5_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.5.1/Ex3.5_1.r
new file mode 100644
index 00000000..a3f3cf30
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.5.1/Ex3.5_1.r
@@ -0,0 +1,16 @@
+# 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 : 3.5     Page No: 170
+# Columns of given matrix are dependent
+library(pracma)
+A<-matrix(c(1,2,1,0,1,0,3,5,3),nrow=3)
+if(Rank(A)!=ncol(A)){
+  print("The columns of A are dependent")
+}
+print("As the columns of A are dependent ,there is a nonzero solution to Ax=0")
+x<-matrix(c(-3,1,1),ncol=1)
+print(A%*%x)
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.1/Ex3.6_1.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.1/Ex3.6_1.r
new file mode 100644
index 00000000..6e7450fc
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.1/Ex3.6_1.r
@@ -0,0 +1,17 @@
+# 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 : 3.6     Page No: 188
+# Dimensions and rank of matrix
+library(pracma)
+solution<-function(A){
+  print(paste("Number of rows , m=",nrow(A)))
+  print(paste("Number of columns,n=",ncol(A)))
+  print(paste("Rank of the given matrix",Rank(A)))
+}
+A<-matrix(c(1,2,3),nrow=1)
+solution(A)
+
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.2/Ex3.6_2.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.2/Ex3.6_2.r
new file mode 100644
index 00000000..71b3fefd
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.2/Ex3.6_2.r
@@ -0,0 +1,16 @@
+# 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 : 2   Chapter : 3.6     Page No: 188
+# Dimensions and rank of matrix
+library(pracma)
+solution<-function(A){
+  print(paste("Number of rows , m=",nrow(A)))
+  print(paste("Number of columns,n=",ncol(A)))
+  print(paste("Rank of the given matrix",Rank(A)))
+}
+A<-matrix(c(1,2,2,4,3,6),nrow=2)
+solution(A)
diff --git a/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.a/Ex3_3.6A.r b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.a/Ex3_3.6A.r
new file mode 100644
index 00000000..9bf248a9
--- /dev/null
+++ b/Introduction_To_Linear_Algebra_by_Gilbert_Strang/CH3/EX3.6.a/Ex3_3.6A.r
@@ -0,0 +1,70 @@
+# 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 : 3.6A   Chapter : 3.6     Page No: 190

+# Four Fundamental Spaces of given 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)

+}

+l<-matrix(c(1,2,5,0,1,0,0,0,1),ncol=3)

+u<-matrix(c(1,0,0,3,0,0,0,1,0,5,6,0),ncol=4)

+A<-l%*%u

+print("Row space Basis of A")

+print(u[1,])

+print(u[2,])

+print("COlumn space Basis of A ")

+print(l[,1])

+print(l[,2])

+print("Null space Basis of A")

+N<-nullspacebasis(A)

+print(N)

+print("Null space Basis of A transpose")

+NT<-nullspacebasis(t(A))

+print(NT)