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\title{MAT 247 \\ Assignment 7 \\ Due Thursday March 17}



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\maketitle
\begin{enumerate}
\item
Consider the polynomial $ p(x,y) = 2x^2 + 4xy + 3y^2$.
\begin{enumerate}
\item
Find a symmetric bilinear form $ H $ on $ \mathbb{R}^2 $ whose associated quadratic form is $ p(x,y) $.
\item
Find a basis for $ \mathbb{R}^2 $ for which this form is diagonal.
\item
Use (b) to decide whether the graph of $ 2x^2 + 4xy + 3y^2 = 1 $ is an ellipse or a hyperbola.
\end{enumerate}

\item Let $ V $ be a vector space.  Let $ v_1, \dots, v_n $ be a basis for $ V $ and let $ \alpha_1, \dots, \alpha_n $ be the corresponding dual basis for $ V^* $.  Let $ w_1, \dots, w_n $ be another basis for $ V $ and let $ \beta_1, \dots, \beta_n $ be the corresponding dual basis.  
\begin{enumerate}
\item Let $ Q $ be the change of basis matrix between $ v_1, \dots, v_n $ and $ w_1, \dots, w_n$.  Show that $ Q^t $ is the change of basis matrix between $ \beta_1, \dots, \beta_n $ and $ \alpha_1, \dots, \alpha_n $.  

(Here by change of basis matrix, I mean that $ w_j = \sum_{i = 1}^n Q_{ij} v_i $.)
\item
Let $ H $ be a bilinear form on $ V $.  Use (a) to prove that 
$$ \left[H \right]_{w_1, \dots, w_n} = Q^t\left[H\right]_{v_1, \dots, v_n}Q  $$
\end{enumerate}
\item
Let $ V, \langle, \rangle $ be a real inner product space.  Let $ T :V \rightarrow V $ be a linear operator.  Let $ H: V \times V \rightarrow \mathbb{F} $ be the map defined by $ H(v,w) = \langle v, Tw \rangle $.
\begin{enumerate}
\item 
Show that $ H $ is a bilinear form on $ V$.
\item
Show that $ H $ is symmetric if and only if $ T $ is self-adjoint.
\item
Show that $H$ is an inner product if and only if $ T $ is positive and invertible.
\end{enumerate}

\end{enumerate}

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