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\newtheorem*{hint}{{\textit{Hint}}}



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\newcommand{\N}{\mathbb{N}} % natural numbers
\newcommand{\R}{\mathbb{R}} % real numbers
\newcommand{\C}{\mathbb{C}} % complex numbers
\newcommand{\Q}{\mathbb{Q}} % rational numbers
\newcommand{\BF}{\mathbb{F}} % field
% \newcommand{\kring}{\mathbf{k}} % commutative ring k (e.g. field)
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
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\newcommand{\onto}{\twoheadrightarrow} % surjection
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% SOME ENVIRONMENTS
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% round matrix
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% square matrix
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\title{MAT 250B HW08}
\author{[add your name here]}
\date{Due Friday, 3/1/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle




%%%%%%%%%%%%%%%%%%%%%%
\exercise

Let $V = \Mat_{2\times 2}(\R)$. Define a form on $V$ by 
\[
    \langle A, B \rangle = \Tr(AB)
\]
where $\Tr$ is the trace. 

\bea

\item Show that this is a symmetric bilinear form.

\item Find the inner product matrix with respect to the standard basis 
\[
    \left \{
    e_{11} = 
    \begin{bmatrix}
        1 & 0 \\ 0 & 0
    \end{bmatrix},
    \quad
    e_{12} = 
    \begin{bmatrix}
        0 & 1 \\ 0 & 0
    \end{bmatrix},
    \quad
    e_{21} = 
    \begin{bmatrix}
        0 & 0 \\ 1 & 0
    \end{bmatrix}, 
    \quad
    e_{22} = 
    \begin{bmatrix}
        0 & 0 \\ 0 & 1
    \end{bmatrix}
    \right \}.
\]
\item Determine the rank and signature of this form.

\ee


%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $V$ be the real vector space consisting of real polynomials of degree $\leq N$. Define a form on $V$ by 
\[
    \langle f,g \rangle = \int_{-1}^1 f(x) g(x) \ dx.
\]
\bea

\item Show that $\langle \cdot, \cdot \rangle$ is a bilinear form on $V$.
\item Prove that $\langle \cdot, \cdot \rangle$ is symmetric and non-degenerate.
\item For $N = 2$, find a basis for $V$ so that the inner product matrix is a diagonal matrix with $\pm 1$ along the diagonal.

\ee



%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $R$ be a commutative ring.

\bea

\item Let $m$ and $n_i$ ($1 \leq i \leq k$) be elements of an $R$-module $M$. 
Prove that 
\[
    m \wedge n_1 \wedge n_2 \wedge \cdots \wedge n_k 
    = (-1)^k n_1 \wedge n_2 \wedge \cdots \wedge n_k  \wedge m.
\]

\item Prove that if $F$ is a free $R$-module of rank $n$, then $\bigwedge^i(F)$ is a free $R$-module of rank ${n \choose i}$. 

\ee

















\end{document}
