1. ## Transition Matrix

Let V = P3 be the vector space of polynomials

a0 + a1x + a2x^2 + a3x^3

og degree smaller than or equal to 3. Let the standard basis be:

E = (1, x , x^2, x^3) and let another basis for V be:

G = (1, (2x + 1), (2x + 1)^2, (2x + 1)^3)

How do I find the transition matrix from G to E?

2. $
E=\left\{
\begin{array}{ccccc}
1,~x,~x^2,~x^3
\end{array}\right\}\rightarrow
[1]_E=\vec{e}_1,~[x]_E=\vec{e}_2,~
[x^2]_E=\vec{e}_1,~ [x^3]_E=\vec{e}_4
$

$
\begin{array}{cccc}
A[1]_E&=A\vec{e}_1&=[1]_E&=\vec{e}_1 \\
A[(2x+1)]_E&=A\vec{e}_2&=[2x+1]_E&=\vec{e}_1+2\vec{e}_2 \\
A[(2x+1)^2]_E&=A\vec{e}_3&=[(2x+1)^2]_E&=\vec{e}_1+4\vec{e}_2+4\vec{e}_3 \\
A[(2x+1)^3]_E&=A\vec{e}_4&=[(2x+1)^3]_E&=\vec{e}_1+6\vec{e}_2+12\vec{e}_3+8\vec{e}_4 \\
\end{array} \\
$

$
\rightarrow A=
\begin{bmatrix}
1&1&1&1 \\
0&2&4&6 \\
0&0&4&12 \\
0&0&0&8 \\
\end{bmatrix}
$

3. Wow, thanks alot! I'm now trying to find the transition matrix from E to G, and using your example I got:

B[1] = [1]E = g1

B[x] = [x]E = -1/2g1 + 1/2g2

B[x^2] = [x^2]E = 1/4g1 - 1/2g2 + 1/4g3

B[x^3] = [x^3]E = -1/8g1 + 3/8g2 - 3/8g3 + 1/8g4

which would give the matrix B =

1,-1/2, 1/4, -1/8
0, 1/2, -1/2,3/8
0, 0, 1/4,-3/8
0, 0, 0, 1/8