If S is the subspace of M5R consisting of all symmetric matr

If S| is the subspace of M_5(R)| consisting of all symmetric matrices, then dim S =| | If S| is the subspace of M_7(R)| consisting of all matrices with trace 0, then dim S =|

Solution

a)

5x5 matrix has 25 entries

5 along diagonal and 25-5=20 away from diagonal. But once entries above the diagonal are determined then the entries below the diagonal are also determined

So ,10 entries above the diagonal only need to be determined

So , 10 +5 along the diagonal =15 total entries need to be determined.

HEnce, dim S=15

b)

Total 49 entries in the matrix. 7 along the diagonal and 42 away

The away entries can vary indepedently.

Along the diagonal determining 6 entries fixes the seventh as trace is 0

HEnce dim S=42+6=48