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Finding a matrix with respect to standard basis

  1. May 28, 2012 #1
    1. The problem statement, all variables and given/known data

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    2. Relevant equations

    ...

    3. The attempt at a solution

    Can someone just point me how to approach this? Do we take a random second degree polynomial and input [itex]2x + 3[/itex] instead of [itex]x[/itex], then find the constants (eg. denoted by [itex]a , b , c[/itex]) by putting the new equation equal to the standard polynomial [itex]1 + x + x^2[/itex]?
     
  2. jcsd
  3. May 28, 2012 #2
    A good start would be to apply ##T## to the basis vectors ##1, x, x^2##. What do you know about ##[T]_B##? In particular, what are its columns?
     
  4. May 28, 2012 #3
    Oh, thank you. This is what I came up with so far:

    [itex]T(1) = 1[/itex] so [itex](T(1))_B = \begin{pmatrix}1\\0\\0\end{pmatrix}[/itex]
    [itex]T(x) = 2x + 3[/itex] so [itex](T(x))_B = \begin{pmatrix}3\\2\\0\end{pmatrix}[/itex]
    [itex]T(x^2) = 4x^2 + 12x + 9[/itex] so [itex](T(x^2))_B = \begin{pmatrix}9\\12\\4\end{pmatrix}[/itex]

    So, [itex]A = [T]_B = \begin{pmatrix}1 & 3 & 9\\0 & 2 & 12\\0 & 0 & 4\end{pmatrix}[/itex]. Is this close?
     
  5. May 28, 2012 #4

    HallsofIvy

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    Better than close- that's exactly what I get.
     
  6. May 28, 2012 #5
    Great! Now may I add a couple more questions to this?

    Part b) asks to find the eigenvalues of [itex]A[/itex]. The values I found were [itex]\lambda_1 = 1, \lambda_2 = 2, \lambda_3 = 4[/itex].

    Part c) asked to find the formula of [itex][T(p(x))]_B[/itex] without justification. I am not sure how to handle this question.
     
  7. May 28, 2012 #6
    The eigenvalues look fine.

    What are the B-coordinates of a polynomial ##p(x) = ax^2 + bx +c##? How does the matrix ##[T]_B## transform those coordinates?
     
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