Example of linear transformation

Question

I need an example of a linear transformation $ T:\mathbb{R}^2 \to \mathbb{R}^2$ such that $N(T)=R(T)$

where $N(T)$ is the null space & $R(T)$ is the range space.

Answer 1

The easiest example is $$ T(x,y)=(y,0) $$ here the $R(T)$ is $(x,0)$ where $ x\in\mathbb{R} $

and $N(T)$ be the same also

Answer 2

Hint Choose the standard basis $e_1,e_2$ on $\Bbb{R}^2$. Say you want $N(T) =R(T) = \langle e_1 \rangle$. Then you want $T(e_1) = 0$ and $T(e_2) = \dots$

Answer 3

First of all, note that $T$ cannot be onto, since in that case $N(T)=0\neq \Bbb{R}^2 = R(T)$.

As an example, just take the projection one one copy of $\Bbb R$, namely $$\begin{align*} T': &&\Bbb R^2 &\to \Bbb R^2\\ T': &&(x,y) &\mapsto (x,0) \end{align*} $$ in which case $N(R)=\left<(0,y)\right>\simeq\Bbb R$ and $T(R)=\left<(x,0)\right>\simeq\Bbb R$.