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Thread: direct sums of matrices

  1. #1
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    direct sums of matrices

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

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

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

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

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

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

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

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

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

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