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–Part 1- Question 1

matrixA<-matrix(c(1,-1,3,-5,2,1,5,-9,6,-1,-2,4),nrow=3, byrow=TRUE)
matrixA
##      [,1] [,2] [,3] [,4]
## [1,]    1   -1    3   -5
## [2,]    2    1    5   -9
## [3,]    6   -1   -2    4
rankofA<-qr(matrixA, LAPACK=FALSE)$rank
print('Rank of the Matrix')
## [1] "Rank of the Matrix"
print(rankofA)
## [1] 3

–Part 1- Question 2

print('Transpose of the Matrix')
## [1] "Transpose of the Matrix"
print(t(matrixA))
##      [,1] [,2] [,3]
## [1,]    1    2    6
## [2,]   -1    1   -1
## [3,]    3    5   -2
## [4,]   -5   -9    4

–Part 1- Question 3 - Also refer the paper work

print('atleast write one orthonormal 3D vector space')
## [1] "atleast write one orthonormal 3D vector space"
v1 = c(1,0,0)

v2=c(0,1,0)

v3=c(0,0,1)

dot.v1.v2<-v1%*%v2
dot.v2.v3<-v2%*%v3
dot.v3.v1<-v3%*%v1

print(dot.v1.v2)
##      [,1]
## [1,]    0
print(dot.v2.v3)
##      [,1]
## [1,]    0
print(dot.v3.v1)
##      [,1]
## [1,]    0
norm.v1 <- norm(v1,type="2")
norm.v2 <- norm(v2,type="2")
norm.v3 <- norm(v3,type="2")

norm.v1
## [1] 1
norm.v2
## [1] 1
norm.v3
## [1] 1

–Part 1- Question 5

print('Eigen Values and Eigen Vectors of  Matrix')
## [1] "Eigen Values and Eigen Vectors of  Matrix"
matrixQ5<-matrix(c(5,0,3,0,1,-2,1,2,0),nrow=3, byrow=TRUE)
print(matrixQ5)
##      [,1] [,2] [,3]
## [1,]    5    0    3
## [2,]    0    1   -2
## [3,]    1    2    0
print(eigen(matrixQ5))
## $values
## [1] 5.471264+0.000000i 0.264368+1.742772i 0.264368-1.742772i
## 
## $vectors
##                [,1]                  [,2]                  [,3]
## [1,]  0.98551404+0i  0.2583505-0.2761849i  0.2583505+0.2761849i
## [2,] -0.06924766+0i -0.6725516+0.0000000i -0.6725516+0.0000000i
## [3,]  0.15481227+0i -0.2473752+0.5860519i -0.2473752-0.5860519i

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