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\title{MAT 247 \\ Assignment 6 \\ Due Thursday March 10}



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\begin{enumerate}
\item (Axler 6.29) Let $ T $ be a linear operator on an inner product space.  Prove that the dimension of the range of $ T $ equals the number of non-zero singular values of $ T$.

\item (Axler 6.33)  Let $ T $ be a linear operator on an inner product space $ V $.  Let $ \hat{s} $ denote the smallest singular value of $ T $ and let $ s$ denote the largest singular value of $ T$.  Show that for all $ v \in V $, 
$$
\hat{s}|| v|| \le ||Tv || \le s||v||.
$$
\item 
\begin{enumerate}
\item
Let $ H_1, H_2 $ be two bilinear forms on a vector space $ V $.  Define a map $ H_1 + H_2 : V \times V \rightarrow \mathbb{F} $ by 
$$ (H_1 +H_2)(v, w) = H_1(v,w) + H_2(v,w).$$  
Show that $ H_1 + H_2 $ is a bilinear form.  
\item
Let $ H $ be a bilinear form on $ V $ and let $ a \in \mathbb{F} $.  Define $ aH : V \rightarrow \mathbb{F} $ by 
$$ (aH)(v,w) = aH(v,w).$$
Show that $aH$ is a bilinear form.
\end{enumerate}
Thus the set of all bilinear forms on a vector space forms a vector space.

\item Let $ V=M_2(\mathbb{F}) $ be the vector space of $2\times 2$ matrices over the field $ \mathbb{F}$.  Define a bilinear form $ H $ on $ V $ by 
$$
H(A,B) = \mathrm{trace}(AB).
$$
Compute the matrix for $ H $ with respect to the usual basis for $ M_2(\mathbb{F})$.  Find a basis for $ M_2(\mathbb{F}) $ for which matrix of $ H $ is a diagonal matrix.  (You may assume that $ 1 + 1 \ne 0 $ in $\mathbb{F}$.)
\end{enumerate}

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