Question

Find a basis for the space of symmetric 2×2-matrices. and show the dim

Answer 1

A 2 x 2 matrix A is of the form

a	b
c	d

where a,b,cd are arbitrary numbers.

If A is a symmetric matrix, then A^T = A so that b = c. Then A changes to

a	b
b	d

Now, let M₁ =

1	0
0	0

M₂ =

0	1
1	0

and M₃ =

0	0
0	1

Then any 2 x 2 symmetric matrix can be expressed as a linear combination of M₁,M₂,M₃. Also, these 3 matrices are linearly independent as none of these can either be expressed as a scalar multiple of another of these matrices or a linear combination of the other 2 matrices.

Hence the set { M₁,M₂,M₃} is a basis for the space of symmetric 2×2-matrices. The dimension of the space is 3.