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Math Help - linear algebra

  1. #1
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    linear algebra

    let T be a linear transformation from V to V. suppose V=s(1)+s(2)+,,,+s(k), where each subspace s(i) is invariant under T. if T can be represented on s(i) by the matrix Bi, show that T can be represented on V by the matrix

    [B1 00000000]
    [0 B2 000000]
    [00 B3 00000]
    [ ............... ]
    [ 0000000 Bk]

    i have no idea what to do. please help.
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  2. #2
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    Quote Originally Posted by Kat-M View Post
    let T be a linear transformation from V to V. suppose V=s(1)+s(2)+,,,+s(k), where each subspace s(i) is invariant under T. if T can be represented on s(i) by the matrix Bi, show that T can be represented on V by the matrix

    [B1 00000000]
    [0 B2 000000]
    [00 B3 00000]
    [ ............... ]
    [ 0000000 Bk]

    i have no idea what to do. please help.
    i think you mean V=s(1) \oplus s(2) \oplus \cdots \oplus s(k), right?
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    i think you mean V=s(1) \oplus s(2) \oplus \cdots \oplus s(k), right?
    yes sorry i didnt know how to type \oplus \
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  4. #4
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    it's really easy: for any 1 \leq j \leq k let T_j=T|_{s(j)} and \sigma_j be a basis for s(j) such that [T_j]_{\sigma_j}=B_j. then \sigma=\bigcup_{j=1}^k \sigma_j is a basis for V and [T]_{\sigma} is the block matrix given in your problem.

    note that we keep the order of elements in \sigma as they are in each \sigma_j.
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