Let V be the vector space of real 2 x 2 matrices with inner product(A|B) = tr(B^tA).Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for U^perp where U^{perp}left{A in V |(A|B)=0 forall B in U right}

texelaare 2021-01-17 Answered

Let V be the vector space of real 2 x 2 matrices with inner product
(A|B)=tr(BtA).
Let U be the subspace of V consisting of the symmetric matrices. Find an orthogonal basis for U where U{AV|(A|B)=0BU}

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Expert Answer

Yusuf Keller
Answered 2021-01-18 Author has 90 answers
Step 1
To find an orthogonal basis of U , where U is the subspace consisting of symmetric matrices.
Let AU then
tr(BtA)=0
tr(BtA)=tr(BtA)
=tr(BA), since B is a symmetric matrix 
=tr(AB)
Hence A is a skew symmetric matrix.
Step 2
Now the basis of U is {[1000],[0110],[0001]}
tr([1000],[0110])=tr[0000]=0
tr([0001],[0110])=tr[0000]=0
tr([1000],[0001])=tr[0000]=0
Hence the given basis is an orthogonal basis.
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Jeffrey Jordon
Answered 2022-01-22 Author has 2047 answers

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