1. Help me!

Let V be the usual real vector space of all 3 x 3 real matrices. Define subsets of V by
U = { A is in V : A^T= A } and W = { Ais in V : A^T= -A }
Show that both U and W are subspaces of V.
Show that V = U (direct sum) W and determine the dimensions of U and W.
Anything will help...

2. Originally Posted by ivanov
Let V be the usual real vector space of all 3 x 3 real matrices. Define subsets of V by
U = { A is in V : A^T= A } and W = { Ais in V : A^T= -A }
Show that both U and W are subspaces of V.
Show that V = U (direct sum) W and determine the dimensions of U and W.
You ought not to need any help to show that U and W are subspaces. To show that V=U⊕W, you need to check two things: (1) every element of V can be expressed as the sum of something in U plus something in W; (2) U∩W={0}.

For (1), notice that $\displaystyle A = \tfrac12(A + A^{\textsc{t}}) + \tfrac12(A - A^{\textsc{t}})$. For (2), you shouldn't need a hint.

To find the dimensions of U and W, notice that an n×n matrix has n^2 entries. Therefore dim(U)+dim(W)=dim(V)=n^2. Now think about dim(W). If a matrix is in W then it must have zeros on the main diagonal, and if you know all the elements above the diagonal then the elements below the diagonal are all determined (they must be the negatives of those that you get by reflecting in the main diagonal). So dim(W) is equal to the number of elements above the diagonal.