Homework Help: Showing a set of matrices is a direct sum.

1. Mar 12, 2013

trap101

Let W1 = {A$\in$ MnXn(R)| A = AT} and W2 = {A$\in$ MnXn(R)| A = -AT}

Show that MnXn = W1 (+) W2

where the definition of direct sum is:

V is the direct sum of W1 and W2 in symbols:

V = W1 (+) W2 if:

V = W1 + W2 and
W1 $\cap$ W2 = {0}

Attempt:

I figure I have to show each property individually. So for the first property I tried to do a manipulation:

AT + (-AT) = (A + (-A))T = 0T

Then I added A to each side: A + (A + (-A))T = A + 0T

Did I even show what was needed?

2. Mar 12, 2013

HallsofIvy

No, you appear to be misunderstanding what "W1= {A$\in$ Mxn|A= AT}" and "W2= {A$\in$ Mnxn|A= -AT[/sub]}" mean. Members of V are of the form A+ B where A is in W1 and B is in W2. You cannot use the same matrix, A, for each.

3. Mar 12, 2013

trap101

How's this:

Letting A be a matrix from W1 and B be a matrix from W2:

V = A + B
= AT+ (-BT)
= (A+(-B))T

Can I bring out the -1 and have: (-1)(A+B)T?