Question

Linear algebra transformations (In comments)

Answer 1

Let P_2 be the vector space of all polynomials of degree 2 or less, and define the transformation \[T: P_2 \rightarrow P_3\] by \[T(p(x))=2x^2p''(x)+p'(x), \forall p(x) \in P_2\]
Find the matrix of T with respect to the basis \[B=\left\{ 1,x,2x^2 \right\}\]
I can get as far as: \[A_B = [(T(1))_B | (T(x))_B|(T(2x^2))_B]=[(0)_B|(1)_B|(8x^2+4x)]\]
But the final answer goes from this line to \[\left[\begin{matrix}0 & 1&0 \\ 0 & 0 & 4\\0&0&4\end{matrix}\right]\]
Would someone be able to explain this last step to me? I know it's lengthy, but I would really appreciate it! Thanks so much!

Answer 2

We need to solve the matrix equation

[1 x 2x^2] [a_11 a_12 a_13] = [0 1 4x+8x^2]
[a_21 a_22 a_23]
[a_31 a_32 a_33]

So we get:

a_11 + a_21 x + 2a_31 x^2 = 0
a_12 + a_22 x + 2a_32 x^2 = 1
a_13 + a_23 x + 2a_33 x^2 = 4x + 8x^2

From first equation, a_11 = a_12 = a_13 = 0
From second equation, a_12 = 1, a_22 = a_32 = 0
From third equation, a_13 = 0, a_23 = 4, 2a_33 = 8, or a_33 = 4.

Hence the matrix is

[0 1 0]
[0 0 4]
[0 0 4]

Answer 3

Thanks so much!