# Math Help - direct sums of matrices

1. ## 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.

2. Originally Posted by studentmath92
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

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