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Math Help - block matrix question

  1. #1
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    block matrix question

    there is a space V of dimention n
    T:V->V
    i am givem also the thing in the photo
    prove that there is a basis B for such [T]_B on in the photo.
    where A C D are mxm matrices.

    i thought to find this basis from the eigenvectors of [T]_B
    but i dont know how because each letter there is a metrices by itself
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: block matrix question

    Perhaps you mean: if T:V\to V is an endomorphism, \dim V=n and W a subspace of V with T(W)\subseteq W then, there exists a basis B of V such that [T]_B=\begin{bmatrix}{A}&{C}\\{0}&{D}\end{bmatrix} . If this is the case, then choose B=\{e_1,\ldots,e_r,e_{r+1},\ldots,e_n\} where \{e_1,\ldots,e_r\} is a basis of W . You'll easily find [T]_B .
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  3. #3
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    Re: block matrix question

    ok so we choose some basis c1..cn .
    i know that each column of [T]_B is T(c_i)_B

    what to do now, i got a very abstract thing .and i need to prove that this basis if following the conditions of the question

    ??
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  4. #4
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    Re: block matrix question

    transgalactic, ask yourself this: for the basis Fernando suggested,

    if we write an element of v \in V as a linear combination:

    v = c_1e_1+\dots + c_re_r + c_{r+1}e_{r+1} \dots + c_ne_n

    what happens when c_{r+1} = \dots = c_n = 0?

    (surely this is the case when when we have an element of W).
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  5. #5
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    Re: block matrix question

    Quote Originally Posted by Deveno View Post
    transgalactic, ask yourself this: for the basis Fernando suggested,

    if we write an element of v \in V as a linear combination:

    v = c_1e_1+\dots + c_re_r + c_{r+1}e_{r+1} \dots + c_ne_n

    what happens when c_{r+1} = \dots = c_n = 0?

    (surely this is the case when when we have an element of W).
    in this case v is a combination of e1 .. e_r
    what now
    stil this block matrices are hard to translate
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  6. #6
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    Re: block matrix question

    Quote Originally Posted by FernandoRevilla View Post
    Perhaps you mean: if T:V\to V is an endomorphism, \dim V=n and W a subspace of V with T(W)\subseteq W then, there exists a basis B of V such that [T]_B=\begin{bmatrix}{A}&{C}\\{0}&{D}\end{bmatrix} . If this is the case, then choose B=\{e_1,\ldots,e_r,e_{r+1},\ldots,e_n\} where \{e_1,\ldots,e_r\} is a basis of W . You'll easily find [T]_B .
    we choose some basis of W and expand it to be the basis of V and show that the representation matrices of T on this expanding basis is the one given in thequestion.
    so we are given that every T(w1)..T(wn) is a combination of W basis

    for example for T(w1)=a1w1+a2w2+..+amwm+..+anwn
    ALSO T(w1) is the first column
    so a_m+1=a_m+2..=a_n =0 these coefficients has to be zero for the first m columns of this matrices in order for it to look we were told to.

    so we just say " the coefffients a_m+1 ..a_n have to be zero in order for this basis to make such a representation matrices"
    and that is the solution?
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