Question

Let W1={A|A∈M2x2, A'=A} and W2={A|A∈M2X2,A'=-A} Prove that M2x2=W1+W2 (direct sum)?

Answer 1

See below.

Explanation:

Any square matrix `M` can be decomposed as a sum of a symmetric part `M_s` plus an antisymmetric part `M_a` being

`M_s =1/2(M + M^T)` with `""^T` meaning transposition, and

`M_a = 1/2(M-M^T)` so

`M = M_s+M_a`