Exercise 6

Decide whether each map is an isomorphism. If it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy.

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto ad-bc$$

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}^{4}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto\begin{pmatrix}a+b+c+d\\ a+b+c\\ a+b\\ a\end{pmatrix}$$

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{3}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto c+(d+c)x+(b+a)x^{2}+ax^{3}$$

• $f\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{3}$ given by

  $$\begin{pmatrix}a&b\\ c&d\end{pmatrix}\mapsto c+(d+c)x+(b+a+1)x^{2}+ax^{3}$$