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Math Help - Transition Matrix

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

    <br />
\begin{array}{cccc}<br />
A[1]_E&=A\vec{e}_1&=[1]_E&=\vec{e}_1 \\<br />
A[(2x+1)]_E&=A\vec{e}_2&=[2x+1]_E&=\vec{e}_1+2\vec{e}_2 \\<br />
A[(2x+1)^2]_E&=A\vec{e}_3&=[(2x+1)^2]_E&=\vec{e}_1+4\vec{e}_2+4\vec{e}_3 \\<br />
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  \\<br />
\end{array} \\<br />

    <br />
\rightarrow A=<br />
\begin{bmatrix}<br />
1&1&1&1 \\<br />
0&2&4&6 \\<br />
0&0&4&12 \\<br />
0&0&0&8 \\<br />
\end{bmatrix}<br />
    Last edited by math2009; March 24th 2010 at 04:39 PM.
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  3. #3
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    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
    Last edited by gralla55; March 24th 2010 at 03:11 PM. Reason: nevermind, I found the error!
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