Thread: subspace 3

Prove that the set S1 of symmetric matrices and the set S2 the anti-symmetric matrices nxn are vector subspaces of the Mnxm and that it has Mnxm = S1 + S2

Originally Posted by Apprentice123

Prove that the set S1 of symmetric matrices and the set S2 the anti-symmetric matrices nxn are vector subspaces of the Mnxm and that it has Mnxm = S1 + S2

let $\displaystyle M_n$ be the vector space of all $\displaystyle n \times n$ matrices over the field $\displaystyle F.$ if $\displaystyle A_1, A_2 \in S_1,$ then for any $\displaystyle c_1,c_2 \in F,$ we have: $\displaystyle (c_1A_1+c_2A_2)^{T}=c_1A_1^T + c_2A_2^{T}=c_1A_1 + c_2A_2,$ i.e. $\displaystyle c_1A_1 + c_2A_2 \in S_1.$

so $\displaystyle S_1$ is a vector space. do the same for $\displaystyle S_2.$ to show that $\displaystyle M_n=S_1 + S_2,$ if $\displaystyle \text{char} \ F \neq 2$ of course, let $\displaystyle A \in M_n$ and put: $\displaystyle A_1=\frac{A+A^T}{2}$ and $\displaystyle A_2=\frac{A-A^{T}}{2}.$ see that $\displaystyle A_1 \in S_1, \ A_2 \in S_2,$ and

$\displaystyle A=A_1+A_2,$ which proves that $\displaystyle M_n=S_1+S_2.$ note that since $\displaystyle S_1 \cap S_2 = 0,$ we actually have: $\displaystyle M_n=S_1 \oplus S_2. \ \ \ \square$

Sorry. More you could be more explanatory