How do you prove this linear transformation $ F \in L(M_{2x2},P_{2})$ is an isomorphism?(verify my solution)

Question

Prove that $$F\begin{bmatrix}a&b\\c&0\end{bmatrix}=ax^2+(a+bx)+a+b+c$$ is an isomorphism.

Context: Elementary Linear Algebra Course. Ok, here is what I ve tried by myself.

Please, verify my solution:

This is a Linear transformation ,

$$ F \in L(M_{2x2},P_{2})$$ I consider that the dimensions of the spaces are $$\dim (M_{2\times2}) = \dim (P_{2}) = 3,$$ so proving it , by lemma: $T$ is injective implies that $T$ is surjective therefore $T$ is an isomorphism.

Actually, by theorem T is injective if Nuc(T)={0} Then, $F\begin{bmatrix}a&b\\c&0\end{bmatrix}=$ $0+(0+0)+0+0+0$ so,

$ax^2+(a+bx)+a+b+c$ $=0+(0+0)+0+0+0$

Finally, $a=b=c=0$ as a matter of fact, $F$ is an isomorphism. Is that correct?, suggestions?.

Thanks in advance. Note: I dont need to prove that it is a linear transformation. Just the isomorphism. Please edit my question if you know how to improve it.

Answer 1

You have a typo (I'm 98% sure), where you want to write your function as: $$F\begin{pmatrix} a & b \\ c & 0 \end{pmatrix} = ax^2 + (a+b)x + a+b+c,$$ note the different placement of the parentheses.

You need to be a little careful in specifying your problem. Like make it clear that you are acting on a 3 dimensional subspace of $M^{2 \times 2}$.

The basic idea of your proof is good though. Just a little more detail/clarity throughout would be beneficial. E.g. how do you conclude $a =b=c=0$? This is an easy computation, but the exercise is an elementary one and the details are important in cases like this.

Answer 2

I assume your vector space is over the field of real numbers in this answer.

It is wrong. The domain is not $M_{2 \times 2}$. You write that $\dim (M_{2 \times 2})=3$, which is also wrong. Rather, you must show that

$$F: S \to P_3: \begin{pmatrix}a & b \\ c & 0 \end{pmatrix} \mapsto ax^2 + (a+bx) + a+ b+c$$

is an isomorphism where $S:= \left\{\begin{pmatrix}a & b \\ c & 0 \end{pmatrix}: a,b,c \in \mathbb{R}\right\}$.

Note first that you have to check that $S$ is a vector space to even be able to talk about linear transformations (hint: show it is a subspace of $M_{2 \times 2}$). The same thing for $P_3$.

Then you can proceed like you did..

Also, I highly dislike the notation $P_3$ for the set of polynomials $\{ax^2+bx + c: a,b,c \in \mathbb{R}\}$. I would rather call this $P_2$, but I guess that's personal taste.