subspace 3

**Apprentice123** · Sep 23rd 2008, 03:57 PM

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

**NonCommAlg** · Sep 23rd 2008, 07:09 PM

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

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

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

**Apprentice123** · Sep 24th 2008, 04:53 AM

Sorry. More you could be more explanatory

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