Identify all vectors in $\mathbb{R}^2$ that are taken by $T$ to the same vector $T(a)$.

Question

If = $$ a = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}\in \mathbb{R}^2\;\;\;\;T\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right) = \begin{bmatrix} 1&2\\3&6\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$ Identify all vectors in $\mathbb{R}^2$ that are taken by $T$ to the same vector $T(a)$.

Here's the full question linked here: screenshot of full problem

I figured out part A of the problem and found a basis for both $\ker(T)$ and $\operatorname{Im}(T)$.

$$ \ker(T) = \operatorname{span} \left\langle\begin{bmatrix} 2 \\ 1 \end{bmatrix}\right\rangle $$and$$ $$

$$ \operatorname{im}(T) = \operatorname{span}\left\langle \begin{bmatrix} 1 \\ 3 \end{bmatrix}\right\rangle $$

Both being $1$-dimensional.

I need some help with part B of the question.

Thanks.

Answer 1

$T(a + k\begin{bmatrix}-2\\1\end{bmatrix}) =T(a)+kT(\begin{bmatrix}-2\\1\end{bmatrix}) = T(a)$

Answer 2

Hint:

If $v \in $Ker$(T)$ then

$T(a+v)=T(a)+0=T(a)$

and, if $T(x)=T(a)$ then

$T(x)-T(a)=T(x-a)=0$