Question C25

Define the linear transformation:

\(T : \C^3 \rightarrow \C^2\), \[T \left(\begin{array} {rr} x_{1} \\ x_{2} \\ x_{3} \\ \end{array}\right) = \left[\begin{array} {rr} 2x_{1} - x_{2} + 5x{3} \\ -4x_{1} + 2x{2} - 10x{3} \\ \end{array}\right]\]

Find a basis for the kernal of \(T\), \(K(T)\). Is \(T\) injective?

\(T(x) = 0\)

\[\left[\begin{array} {rr} 2x_{1} - x_{2} + 5x{3} \\ -4x_{1} + 2x{2} - 10x{3} \\ \end{array}\right] = \left[\begin{array} {rr} 0 \\ 0 \\ \end{array}\right]\]

First, set both equations to equal zero.

a <- matrix(c(2, -4, -1, 2, 5, -10, 0, 0), ncol = 4, byrow = F)
a
##      [,1] [,2] [,3] [,4]
## [1,]    2   -1    5    0
## [2,]   -4    2  -10    0
A <-rref(a)
A
##      [,1] [,2] [,3] [,4]
## [1,]    1 -0.5  2.5    0
## [2,]    0  0.0  0.0    0

I know from here I’m meant to find the basis for null space, but I’m totally lost on how to do that. Any help would be so appreciated.