# Compute preimage of linear transformation

I'm working on an assignment and I've pored over my lecture notes, but I can't find anything about this.

I'm told to define the linear transformation $T:\mathbb{C}\rightarrow\mathbb{C}$ by $$T\left(\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}\right)= \begin{bmatrix} 2x_1-x_2+5x_3\\ -4x_1+2x_2-10x_3 \end{bmatrix}.$$

Then I'm asked to compute the preimage of $T^{-1} \left(\begin{bmatrix} 2\\ 3 \end{bmatrix}\right)$, and find a basis for the kernel of $T$ and asked whether $T$ is injective.

Based on this question, Finding the pre-image of a linear transformation, which is very similar, do I simply row reduce $$\begin{bmatrix} 2 & -1 & 5 & 2\\ -4& 2 & -10 & 3\\ \end{bmatrix}?$$

If so, that system is inconsistent. So is there no preimage?

• yes and yes. {}{} – qbert May 9 '17 at 0:03
• Sure looks that way. – amd May 9 '17 at 0:05
• Alrighty, thanks, @qbert and @amd! :) – jbrow35 May 9 '17 at 0:07
• The correct spelling is "pore over", by the way. – Omnomnomnom May 9 '17 at 13:30
• @Omnomnomnom You're absolutely right, how embarrassing! – jbrow35 May 9 '17 at 14:57