Determining if a linear transformation is one to one

Question

$T$ is the linear transformation from $\mathbb{R}^3 $ to $ \mathbb{R}^3$ : $T(a1, a2, a3) = (a1-a2, a2-a3, a3-a1)$

Question: Is $T$ one to one ?

Attempt:

After row reducing , I get that $a1-a3 = 0$ , $a2-a3 = 0$. $a3$ is free so no its not one to one.

Answer 1

Yes it is not one to one indeed the matrix associated is

$$[T]=\begin{bmatrix}1&-1&0\\0&1&-1\\-1&0&1 \end{bmatrix}$$

with $rank(T)=2\implies \det(T)=0$ thus $[T]$ is not invertible.