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Math Help - Computing Linear operator on the basis B.

  1. #1
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    Computing Linear operator on the basis B.

    Compute [T]_B and determine whether B is a basis consisting of eigenvectors of T.



    a) V= P_2(R), T(a+bx+cx^2)= (-4a+2b-2c)-(7a+3b+7c)x +(7a+b+5c)x^2,

    and a basis B= {x-x^2, -1+X^2, -1-x+x^2}

    b)V=M_2x2(R), \begin{pmatrix}\ a & b \\ c & d\end{pmatrix}\


    = \begin{pmatrix}\-7a-4b+4c-4d & b\\-8a-4b+5c-4d & d\end{pmatrix}\


    And the basis B = { \begin{pmatrix}\ 1 & 0 \\ 1 & 0\end{pmatrix} \begin{pmatrix}\ -1 & 2 \\ 0 & 0\end{pmatrix} \begin{pmatrix}\ 1 & 0 \\ 2 & 0\end{pmatrix} \begin{pmatrix}\ -1 & 0 \\ 0 & 2\end{pmatrix}}
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  2. #2
    Senior Member
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    I am going to call the standard basis for P_2(\mathbb{R})\quad E for no good reason

    [T]_B = [E\to B][T][B \to E]
    Where [E \to B] is the matrix that converts from basis E to B. We can get the matrix [B \to E] by using the basis vectors as columns. The matrix [E\to B] is simply the inverse of [B \to E].
    B is a basis consisting of eigenvectors of T iff [T]_B is diagonal.

    go for it.
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