If S is the subspace of Ropf6 times 6 consisting of all symm

If S is the subspace of Ropf^6 times 6 consisting of all symmetric matrices, then dim s = _____ If S is the subspace of Ropf^3 times 3 consisting of all matrices with trace 0, then dim s = ___

Solution

a)

6x6 matrix have 36 entries

But if matrix is symmetrix then determining entries above the diagonal determines entries below the diagonal

Entries above teh diagonal are: (36-6)/2=15 in number

So total number of entresi which can be filled independently are :15+6 diagonal entries =21

Hence, dim S=21

b)

3 entries along the diagonal and 6 away from teh diagoanl

Trace=0 only puts restriction on diagonal entries

and determining two diagonal entries fixes teh third since trace is 0

SO, dim S=6+2=8