Linear Algebra, how to solve transformation T: ℙ2→M 2,2?

Question

Could someone please help me with two questions?

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So I know they must be a basis of P3, so

p = a + bx + cx^2 + dx^3

I then need to find a way to sub the equation inside but I have no idea how to do so. I am using the Lyryx textbook, which did not really explain any of the steps so if anyone could teach me step by step, that would be great help!

Thank you!

Answer 1

First one: $2q-p=r\implies T(r)=T(2p-q)=2T(p)-T(q)=\begin{pmatrix}5&11\\-21&0\end{pmatrix}$.

Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.

Multiply through and set the coefficients equal. Solve the system.

I get: $a_1=a+c-d\\a_2=b+c-d\\a_3=2c-d \\a_4=-c+d$.

Now use the matrices given and the linear property. I get

$\begin{pmatrix}9a+3b-3d&-6a+3b\\-5a-b+2c-6d&4a-b-3c+5d\end{pmatrix}$.