I need to construct an isomorphism:
$T: W \to C^3$
where W is a subspace of $P_3(C)$
I got bases (for W and $C^3$ respectively)
$\alpha = \{(ix + 1),\ \ (x^3 + 2x), \ (2ix^2 + ix -1)\}$
$\beta = \{ (1,0,0),\ (0,1,0),\ (0,0,1)\}$
obviously
$[T]_{\beta\alpha} = \matrix{1\ \ \ 0\ \ \ 0\\0\ \ \ 1\ \ \ 0\\0\ \ \ 0\ \ \ 1\\}$
but how do I find an explicit formula T(p(x))?
Thank you!
$\endgroup$1 Answer
$\begingroup$The map $T$ sends $a(ix+1)+b(x^3+2x)+c(2ix^2+ix-1)$ onto $(a,b,c)$. If you want something more explicit, you can write
$$a(ix+1)+b(x^3+2x)+c(2ix^2+ix-1)=bx^3+2ic \cdot x^2+(2b+ic)\cdot x+(a-c)$$
So your map sends a polynomial $\alpha x^3+\beta x^2+\gamma x+\delta$ onto $(\delta+\beta/(2i),\alpha,\beta/(2i))$.
(Of course not all polynomials belong to your subspace).
$\endgroup$ 3
