Data 605 - Week 4 Discussion

knitr::include_graphics('Week4.png')

# Creating matrix into S
S <- matrix(c(-1,4,1,1,4,-1,5,0,1,6,-2,2,2,-1,2,5), 4,4)
S

##      [,1] [,2] [,3] [,4]
## [1,]   -1    4    1    2
## [2,]    4   -1    6   -1
## [3,]    1    5   -2    2
## [4,]    1    0    2    5

# RREF
library(pracma)
answer <- rref(S)
answer

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1

The identity matrix is nonsingular and have null space therefore it is injective. U’s dimension is 4 and kernel is 0 therefore it is also surjective. Lastly, transformation is injective and surjective therefore it is also invertible as well as per ILTIS theorem.