Determine whether the transformation is an isomorphism

Question

I have a linear transformation $Q : P_5 \to M_{2,3}$, given by

$$Q(a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5) = \begin{bmatrix}a_0& a_1& a_2\\ a_5-a_4&a_5+a_4& a_3 \end{bmatrix}.$$

So I know that $Q$ has to be a one-to-one linear transformation of $P$ onto $M$ but where do I go from there? Is it not isomorphism because of $P_5$?

Answer 1

In terms of the standard bases, the matrix of $Q$ is

$$\begin{pmatrix} 1 & 0 &0 & 0& 0&0 \\ 0 & 1 &0 & 0& 0&0 \\ 0 & 0& 1& 0& 0& 0\\ 0 &0 & 0& 0& -1& 1\\ 0 & 0& 0& 0& 1& 1\\ 0 & 0& 0& 1& 0& 0\\ \end{pmatrix}.$$

$Q$ represents an isomorphism iff this matrix is invertible

iff it has full rank

iff its null space is trivial

iff its determinant is nonzero... choose your preferred method.