How can I find Linear transformation $T: \mathbb R_2 [x] \to M_2(\mathbb R)$ so that $Ker$ $T=0$?

Question

How can I find Linear transformation $T: \mathbb R_2 [x] \to M_2(\mathbb R)$ so that $Ker$ $T=0$?

I don't want a final solution, I want explain how can I make example like that? what is the algorithm?

Answer 1

$\mathbb R_2 [x]$ has dimension 3, $M_2(\mathbb R)$ has dimension 4. A linear map has $Ker = 0$ iff is injective. Pick a base of your choice of the first vector space and send each vector to a different element of a base of the second vector space of your choice.

A basis for $\mathbb R_2 [x]$ is $1, x, x^2$. A basis for $M_2(\mathbb R)$ is $E_{1,1}, E_{1,2}, E_{2,1}, E_{2,2}$, where $E_{i,j}$ is the matrix with all $0$ except the entry $(i,j)$ which is $1$. We can map for example $1 \to E_{1,1}$, $x \to E_{1,2}$, $x^2 \to E_{2,1}$. If you extend linearly this map to the whole $\mathbb R_2 [x]$ you get a linear map whose $Ker$ is $0$.