3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – Michael Hardy Apr 22 '13 at 17:07
  • $\begingroup$ Thank you for that, I'm not too good at the TeX commands. $\endgroup$ – Dexter Apr 22 '13 at 17:08
1
$\begingroup$

Yes, your answers are correct.

$\endgroup$
  • $\begingroup$ Thank you, when I say "they are linearly independent" about the set of matrices, do I need to add "and also spanning" or not? $\endgroup$ – Dexter Apr 22 '13 at 17:03
  • $\begingroup$ That needs to be true for them to be a basis, but the way you've phrased that sentence it's clear that you are using that they span. $\endgroup$ – Jim Apr 22 '13 at 17:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.