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\begin{document}
\title{Final Week Worksheet}
\author{Renzo's math 469}
\date{}

\maketitle

{\it
This worksheet constitutes the in class/take home part of your final exam. You are encouraged to work collaboratively, and to ask questions to  Jess and Rachel. In the end, please write up your own excellent, tidy, beautifully presented and eloquent solutions for $\mathbf{6}$ of the problems and turn them in to me electronically or physically by Tuesday 5/10.
}


\section*{The determinant}

Recall that given a vector space $V$ with a basis $e_1, \ldots e_n$, the $k$-th alternating product $\Lambda^k(V)$ is a vector space of dimension ${{n}\choose{k}}$ with basis 
given by all symbols $e_{i_1}\wedge \ldots \wedge e_{i_k}$ with $i_1 <i_2 <\ldots ,i_k$. 

If $T:V \to W$ is a linear function, then $\Lambda^k(T): \Lambda^k(V) \to \Lambda^k(W)$ is defined on decomposable tensors by:
$$
\Lambda^k(T) (v_1 \wedge \ldots \wedge v_n) =T(v_1) \wedge \ldots \wedge T(v_n)
$$
and extended to arbitrary linear combinations of tensors by linearity.


\begin{p}
Let $V$ be a four dimensional vector space with bases $e_1, e_2, e_3 , e_4 $, and  $W$ a  three dimensional vector space with basis $f_1, f_2, f_3$.  Let $T: V\to W$ be the linear transformation defined by:
$$
T(e_1) = f_1 + 2 f_3  \ \ \ \ \ T(e_2) = 2f_2 +  f_3  \ \ \ \ \ T(e_3) = 3f_1 + 4 f_2  \ \ \ \ \ T(e_4) = f_1 +  f_3   
$$ 

Write a matrix representing the linear transformation 
$$
\Lambda^2(T): \Lambda^2(V) \to \Lambda^2(W).
$$
\end{p}

Recall the axioms of a functor.

A {\bf covariant functor} from the category of vector spaces to itself is an assignment $\mathfrak{F}$ such that:

\begin{enumerate}
\item  For any vector space $V$, $\mathfrak{F}(V)$ is a vector space.
\item For any linear map $T: V\to W$,  $\mathfrak{F}(T)$
is a linear map $\mathfrak{F}(T): \mathfrak{F}(V)\to \mathfrak{F}(W)$.
\item For any $V$, \[\mathfrak{F}(Id_V)= Id_{\mathfrak{F}(V)}\]
\item For any pair of linear functions:
\[
U \stackrel{T}{\to}V \stackrel{S}{\to}W,
\]
we have that
\[
\mathfrak{F}(S\circ T) = \mathfrak{F}(S) \circ \mathfrak{F}(T)
\]
\end{enumerate}


\begin{p}
Prove that for any positive integer $k$, the $k$-th alternating product
$\Lambda^k$ is a covariant functor from the category of vector spaces to the category of vector spaces.
\end{p}


\begin{p}
If $V$ is a $k$-vector space of dimension $n$, prove that $\Lambda^n(V) \cong k$, and $\Lambda^k(V) =0$ for  $k>n$. 
\end{p}

\begin{p}
Let $T \in \mathcal{L}(\bR^2)$ be represented by the matrix 
$$
M(T)=
\left[
\begin{array}{cc}
a &  b \\
c  & d
\end{array}
\right]
$$ 
Describe the linear transformation $\Lambda^2(T)$.
\end{p} 

\begin{defi}
Let $V$ be a vector space of dimension $n$ and  $T\in \mathcal{L}(V)$. Then $\Lambda^n(T):\Lambda^n(V) \cong k \to \Lambda^n(V) \cong k$ must be multiplication by a scalar factor, 
which we define to be  the {\bf determinant} of $T$.
\end{defi}



\begin{p}
Compute the determinant of the following linear transformations:
\begin{itemize}
	\item $T$ defined by $T(e_1) =  e_1+2 e_2, T(e_2) =  3e_1+ e_3, T(e_3) =  5 e_2 -e_3$;
	\item $F$ defined by $F(x,y,z) = (x+3z, 2x-y, y+7z)$;
	\item $U$ represented by the matrix
$$
M(U)=
\left[
\begin{array}{ccc}
2 &  1 &  0\\
0 &  1 &  -2\\
3 &  0  &  3
\end{array}
\right]
$$
\end{itemize}
\end{p}

\begin{p}
Show that the determinant of the identity function $Id. : V\to V$  is equal to $1$.
\end{p}

\begin{p}
Prove that the determinant  of a composition of operators equals the product of determinants. If $S, T\in \mathcal{L}(V)$:
$$
det(S\circ T) = det(S) det(T)
$$
\end{p}

\begin{p}
Prove that if $V$ is an $n$-dimensional vector space and $T\in \mathcal{L}(V), \lambda \in k$:
$$
det(\lambda T) = \lambda^n det(T)
$$
\end{p} 

\begin{p}
Show that in general 
$$
(v_1+w_1) \wedge ( v_2+w_2) \wedge \ldots \wedge (v_n+w_n) \not= (v_1 \wedge  v_2 \wedge \ldots \wedge v_n)+(w_1 \wedge  w_2 \wedge \ldots \wedge w_n)
$$
and explain how this is the reason for the determinant not being additive:
$$
det(T+S)\not= det(T) +det(S)
$$
\end{p}

\begin{p}
If the matrices $A_1$ and $A_2$ are related by interchanging two columns, show that
$$ det(A_1) = -det(A_2)$$  
\end{p}

\begin{p}
Show that if a column in a matrix $A$ is a linear combination of the other columns then 
$$
det(A)=0
$$
\end{p}

\begin{p}
For $T\in \mathcal{L}(V)$, show that
$$
det(T) = 0 \iff T \mbox{\ \ is not invertible.}
$$
\end{p}

\begin{p}
Let $A= [a^j_i]$ be an upper (or lower) triangular matrix. Prove that the determinant of $A$ is the product of the diagonal entries
$$
det(A) = \prod_i a_i^i
$$
\end{p}

\begin{p}
Let $V$ be a complex vector space over $\mathbb{C}$, and $T\in \mathcal{L}(V)$. Prove that $\varphi(\lambda) =det(T-\lambda Id.)$ is the characteristic polynomal of $T$. Show that $det(T)$ is the constant coefficient of the characteristic polynomial. 
\end{p}


\begin{p}
Given a matrix $A$, prove that 
$$
det(A) = det(A^T)
$$
as follows. Assume that $A$ represents a linear operator $T$ on  $V$ a complex vector space (You can assume that the transformation is on a complex vector space even if  $A$ has real entries). Then you can choose a basis such that...
\end{p}

\begin{p}
Let $V$ be a real vector space with an inner product. Show that an isometry $S\in \mathcal{L}(V)$ has determinant $\pm 1$ 
\end{p}

\begin{p}
Consider $\bR^2$ with the standard inner product, and standard orthonormal basis $e_1, e_2$. Show that for any pair of vectors $v, w$,
$$
v \wedge w =  \alpha (e_1 \wedge e_2)
$$
where $|\alpha|$ is the area of the parallelogram determined by $v$ and $w$.  Prove (I suggest by induction) that this statement generalizes to arbitrary dimension, by replacing parallelogram with $n$-dimensional parallelepiped and area by $n$-dimensional volume.
\end{p}

\begin{p}
Let $V = \bR^n$ with the standard inner product, and $T\in \mathcal{L}(V)$. The number $det(T)$ is the constant of proportionality by which the volume of any shape in $V$ is scaled by $T$.
Don't show this in full generality though (to do so requires integration and all that jazz, and we are not focusing on that), but prove it for  parallelepipeds of the form $v_1 \wedge \ldots \wedge v_n$. 
\end{p}
 

\end{document}