# Thread: direct sums of matrices

1. ## 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.

2. Originally Posted by studentmath92
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

3. Originally Posted by aman_cc
$\displaystyle W_1$ $\displaystyle \oplus$ $\displaystyle W_2$ 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.