605D4

Exercise ILTC27

Let T: C^3 -> C^3 be given by T(c(X,Y,Z)) =

2x + y + z
x - y + 2z
x + 2y - z

Find K(T)

Solution

if T(c(X,Y,Z)) =

2x + y + z
x - y + 2z
x + 2y - z

= 0

Then,

2x+y+z = 0
x-y+2z = 0
x+2y-z = 0

library(Matrix)
library(pracma)

## Warning: package 'pracma' was built under R version 3.6.3

## 
## Attaching package: 'pracma'

## The following objects are masked from 'package:Matrix':
## 
##     expm, lu, tril, triu

M <- matrix(c(2,1,1,1,-1,2,1,2,-1),nrow = 3, byrow = TRUE)

rref(M)

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

x = -z y = z

K(T) = c(-1,1,1)

T is not injective