Question

Given the linear transformation below:

T (x₁, x₂, x₃) → (x₁-x₂+2x₃, 2x₁-2x₃, -x₁-x₂+4x₃, 3x₁-x₂)

T = R³ → R⁴

1. Determine the transformation matrix of the linear transformation above

2. Determine Ker(T) and Range(T)

Answer 1

1.

$Ax=\begin{pmatrix} 1 & -1&2 \\ 2&0 & -2\\ -1&-1&4\\ 3&-1&0 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3\\ \end{pmatrix}=\begin{pmatrix} x_1-x_2+2x_3 \\ 2x_1-2x_3 \\ -x_1-x_2+4x_3\\ 3x_1-x_2 \end{pmatrix}$

transformation matrix:

$A=\begin{pmatrix} 1 & -1&2 \\ 2&0 & -2\\ -1&-1&4\\ 3&-1&0 \end{pmatrix}$

2.

kernel:

$x_1-x_2+2x_3=0$

$2x_1-2x_3=0$

$-x_1-x_2+4x_3=0$

$3x_1-x_2=0$

$x_1=x_3,x_2=3x_1$

$ker\ T=span(1,3,1)$

range:

$range\ T=\begin{pmatrix} x_1-x_2+2x_3 \\ 2x_1-2x_3 \\ -x_1-x_2+4x_3\\ 3x_1-x_2 \end{pmatrix}=x_1\begin{pmatrix} 1 \\ 2 \\ -1\\ 3 \end{pmatrix}+x_2\begin{pmatrix} -1 \\ 0 \\ -1\\ -1 \end{pmatrix}+x_3\begin{pmatrix} 2 \\ -2 \\ 4\\ 0 \end{pmatrix}=$

$=span((1,2,-1,3),(-1,0,-1,-1),(2,-2,4,0))$