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\title{Problem Set 6}
\author{Matrix Theory \& Linear Algebra II}
\date{Winter 2025}
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\noindent
In this problem set, any vector space $V$ comes equipped with an inner product.
\begin{enumerate}[(1)]
    \item 
    Choose one of the inner products defined in Example 6.3 of the book.
    Check that they are indeed inner products, i.e. show that the axioms are satisfied.
    \item
    Let $e_1,\dots,e_n$ be an orthonormal basis, and $v=a_1e_1+\dots+a_nv_n$.
    Show that $a_i = \langle v,e_i\rangle$.
    \item 
    Suppose that $u,v\in V$ and $\|u\|=\|v\|=1$ and $\langle u,v\rangle =1$.
    Prove that $u=v$.
    \item 
    Recall that in $\mathbb R^3$ it's true that
    \[
    \vec u\cdot \vec v = \|\vec u\|\|\vec v\|\cos\alpha,
    \]
    where $\alpha$ is the angle between the vectors.
    There is no ``angle'' between vectors in a general vector space $V$, but in the presence of an inner product we can use this formula to define the angle between two vectors:
    \[
        \cos(u,v) =\arccos\left(\dfrac{\langle u,v\rangle}{\|u\|\|v\|}\right).
    \]
    \begin{enumerate}[(a)]
    \item 
    Show that this formula is well-defined.
    (What is the domain of $\arccos$?)
    \item 
    Using the inner product $\langle f,g\rangle=\int_0^1 f(x)g(x)dx$ on $C^0([0,1])$, calculate the angle between $x$, $e^x$, and $\sin x$. 
    \end{enumerate}
    \item 
    Let $\mathcal B = e_1,\dots,e_n$ be an orthonormal basis for $V$, and define the matrix $A$ whose entries are $a_{ij} = \langle e_i,e_j\rangle$.
    Show that, for $u,v\in V$,
    \[
    \langle u,v\rangle = [u]_\mathcal B^\intercal \cdot A \cdot \overline{[v]}_\mathcal B,
    \]
    where $\overline{[v]}_\mathcal B$ denotes the vector with the conjugate of coordinates of $v$ as its entries.

    \item 
    In $\mathbb R^3$ with the usual dot product, find an orthonormal basis for
    \[
    U = \text{span}\left(\begin{bmatrix}
        1\\2\\3
    \end{bmatrix},\begin{bmatrix}
        2\\6\\0
    \end{bmatrix}\right).
    \]

    \item 
    Consider the inner product space $C([0,2])$ with the inner product given by 
    \[
    \langle f,g\rangle = \int_0^2f(x)g(x)dx.
    \]
    Use Gram-Schmidt  to find an orthogonal basis for $\{1,x,x^2\}$.

    \item 
    Consider the vector space $V=C([0,2\pi],\mathbb C)$ of continuous functions $f:[0,2\pi]\to\mathbb C$.
    An element of $V$ is a function of the form \[f(x) = u(x)+iv(x),\] where $u$ and $v$ are real valued functions $[0,2\pi]\to \mathbb R$.
    Suppose $f=u+iv$ and $g=s+it$.
    Define an inner product of $V$ by
    \[
    \langle f,g\rangle
    =\int_0^{2\pi} u(x)s(x)dx+i\int_0^{2\pi}v(x)t(x)dx.
    \]
    Calculate the inner product between the functions $e^{ix}=\sin x+i\cos x$, $1$, and $x$.
    \item 
    In $\mathbb C^3$ with the complex dot product, find an orthonormal basis for
    \[
    U = \text{span}\left(\begin{bmatrix}
        0\\i\\2
    \end{bmatrix},\begin{bmatrix}
        1\\i+1\\3i+2
    \end{bmatrix}\right).
    \]
    
    
\end{enumerate}

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