# 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}

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

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Yusuf Keller
Step 1
To find an orthogonal basis of ${U}^{\perp }$ , where U is the subspace consisting of symmetric matrices.
Let $A\in {U}^{\perp }$ then
$tr\left({B}^{t}A\right)=0$
$⇒tr\left({B}^{t}A\right)=-tr\left({B}^{t}A\right)$

$=tr\left(-AB\right)$
Hence A is a skew symmetric matrix.
Step 2
Now the basis of ${U}^{\perp }$ is $\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$
$tr\left(\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]\right)=tr\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$
$tr\left(\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]\right)=tr\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$
$tr\left(\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right)=tr\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=0$
Hence the given basis is an orthogonal basis.
Jeffrey Jordon