# Find the basis for $\text{Im} \, ψ$ of a matrix transformation

Let $\psi\colon\mathrm{Mat}_{ 2\times 2 }(\mathbb R) \to \mathrm{Mat}_{ 2\times 2 }(\mathbb R)$ be defined by

$$\psi\colon \pmatrix{a&b\\c&d}\to\pmatrix{a+b&a-c\\a+c&b-c}.$$

Find basis for $\operatorname{Im}\psi$ (image of $\psi$) .

Now I have got that the $\operatorname{Im}\psi$ is $\operatorname{span}\left\{ \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) % ,\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \ % ,\left(\begin{array}{cc} 0 & -1 \\ 0 & -1 \end{array} \right)\right\}$.

And that the basis is the set of these matrices: $\left\{ \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right) % ,\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \ % ,\left(\begin{array}{cc} 0 & -1 \\ 0 & -1 \end{array} \right)\right\}$ since they are linearly independent.

Am I right about the $\operatorname{Im}\psi$ and it's basis?

• I changed $2x2$ to $2\times 2$, and {$\begin{bmatrix} \bullet \\ \bullet\end{bmatrix}$} (with curly braces outside the math environment) to $\left\{\begin{bmatrix} \bullet \\ \bullet\end{bmatrix}\right\}$ (with the braces inside the math environment), and did some other $\TeX$ improvements. Putting actual text inside the math environment without using \text{} is not considered correct, and in this case it's better just to put it outside the math environment. – Michael Hardy Apr 22 '13 at 17:07
• Thank you for that, I'm not too good at the TeX commands. – Dexter Apr 22 '13 at 17:08