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Math Help - subspace 3

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by Apprentice123 View Post
    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
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  3. #3
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    Sorry. More you could be more explanatory
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