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Math Help - direct sums of matrices

  1. #1
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    direct sums of matrices

    In M_{mxn}(F) define  W_1 = {A  \in M_{mxn}(F): A_{ij} = 0 whenever i > j} and W_2 = {A  \in M_{mxn}(F): A_{ij} = 0 whenever i \leq j}. Show that M_{mxn}(F) = W_1 \oplus W_2.
     W_1 is the set of all upper triangular matrices defined as follows: An  {m}x {n} matrix W is called upper triangular if all entries lying below the diagonal entries are zero, that is, if W_{ij} = 0 whenever i >j.
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  2. #2
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    Quote Originally Posted by studentmath92 View Post
    In M_{mxn}(F) define  W_1 = {A  \in M_{mxn}(F): A_{ij} = 0 whenever i > j} and W_2 = {A  \in M_{mxn}(F): A_{ij} = 0 whenever i \leq j}. Show that M_{mxn}(F) = W_1 \oplus W_2.
     W_1 is the set of all upper triangular matrices defined as follows: An  {m}x {n} matrix W is called upper triangular if all entries lying below the diagonal entries are zero, that is, if W_{ij} = 0 whenever i >j.

    W_1 \oplus W_2 will be a sub-space of M_{m\times n}

    consider any x \in M_{m\times n}

    x can be written as w1 + w2 where w1 \in W_1 and w2 \in W_2

    Thus x \in M_{m\times n} => x \in W_1 \oplus W_2

    Hence the result
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  3. #3
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    Quote Originally Posted by aman_cc View Post
    W_1 \oplus W_2 will be a sub-space of M_{m\times n}

    consider any x \in M_{m\times n}

    x can be written as w1 + w2 where w1 \in W_1 and w2 \in W_2

    Thus x \in M_{m\times n} => x \in W_1 \oplus W_2

    Hence the result
    to prove the sum is "direct" you also need to show that W_1 \cap W_2 = \{0 \}, which, of course, is trivial.
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