Say whether $V = \mathcal{P}_{3}(\mathbb{R})$ is isomorphic to $W = M_{2\times 2}(\mathbb{R})$ and if it is give an isomorphism.

Question

Since the dimensions are same I am assuming an isomorphism exists. I know I would have to check whether it is one-to-one and onto, but how to I give an isomorphism. I am confused on what that means. Thank you in advance!

Answer 1

HINT

Since they have the same dimension, they are isomorphic.

In order to conclude so, we shall proceed as follows.

To begin with, consider the following ordered basis for $V$: \begin{align*} \mathcal{B}_{V} = \{v_{1},v_{2},v_{3},v_{4}\} = \{1,x,x^{2},x^{3}\} \end{align*} Similarly, consider the following ordered basis for $W$: \begin{align*} \mathcal{B}_{W} = \{w_{1},w_{2},w_{3},w_{4}\} = \left\{ \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1\\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix} \right\} \end{align*}

Then there exists (and it is unique!) the linear transformation $T:V\to W$ defined by $T(v_{i}) = w_{i}$.

Can you prove that $T$ is injective in order to conclude that $T$ is an isomorphism based on the rank-nullity theorem?