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\title{MAT 247 \\ Assignment 1 \\ Due Thursday January 20}



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Throughout the assignment, use the convention that $ \langle a v, w\rangle = \bar{a} \langle v, w \rangle $ and $ \langle v, aw \rangle = a \langle v, w \rangle$.

\begin{enumerate}
\item
Let $ V $ an inner product space and let $ v, w \in V $.  Prove that $ \langle v, w \rangle = 0 $ if and only if $ ||v || \le || v + a w || $ for all $ a \in \mathbb{F} $ (here $\mathbb{F} = \mathbb{C} $ or $\mathbb{R}$).

\item
On $ \mathcal{P}_2(\mathbb{R}) $ (the vector space of polynomials of degree less than or equal to 2), consider the inner product given by
$$
\langle p, q \rangle = \int_0^1 p(x) q(x) \, dx.
$$
Apply the Gram-Schmidt procedure to the basis $ (1, x, x^2) $ to produce an orthonormal basis of $ \mathcal{P}_2(\mathbb{R}) $.

\item
What happens if the Gram-Schmidt procedure is applied to a list of vectors that is not linearly independent?

\item
Let $ V, \langle, \rangle $ be a finite dimensional real inner product space.

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\item
For each $ v \in V$, define a map $ T_v : V \rightarrow \mathbb{R} $ by $$ T_v(w) = \langle w, v \rangle. $$

Show that for each $ v \in V $, $ T_v $ is a linear map.

\item
Now define a map $ F : V \rightarrow V^* $ by $ F(v) = T_v $.  Here $ V^* = L(V, \mathbb{R}) $ denotes the dual space of $ V $.  Show that $ F $ is a linear map.

\item
Prove that $ F $ is invertible.  (First prove that it is injective and then use that $ \dim V = \dim V^* $).

\item
What happens if $ V, \langle, \rangle $ is a complex inner product space?

\end{enumerate}

\end{enumerate}

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