Let V be a vector space of all 6x6 real matrices over the real field R. Then dimension of subspace of V consisting of all the symmetric matrices.

Question

Options

a) 21

b) 18

c) 15

d) 10

Answer 1

The answer is 15.

I don't think this thing supports LaTeX so I will try to just describe a basis. Label the entries of the matricies by (i ,j) for (row, column). The following 15 matricies form a basis:

1) Entries are all 0 except for a 1 in (1,1). (That is, the entry in the first row, first column should be 1)

2) Entries are all 0 except for a 1 in (2,2).

3) Entries are all 0 except for a 1 in (3,3).

4) Entries are all 0 except for a 1 in (4,4).

5) Entries are all 0 except for a 1 in (5,5).

6) Entries are all 0 except for a 1 in (6,6).

7) Entries are all 0 except for 1s in (1,2) and (2,1).

8) Entries are all 0 except for 1s in (1,3) and (3,1).

9) Entries are all 0 except for 1s in (1,4), (2,3), (3,2), and (4,1).

10) Entries are all 0 except for 1s in (1,5), (2,4), (4,2), and (5,1).

11) Entries are all 0 except for 1s in (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

12) Entries are all 0 except for 1s in (2,6), (3,5), (5,3), and (6,2).

13) Entries are all 0 except for 1s in (3,6), (4,5), (5,4), and (6,3).

14) Entries are all 0 except for 1s in (4,6) and (6,4).

15) Entries are all 0 except for 1s in (5,6) and (6,5)