Midterm 2 Review Problems

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Apologies if you can't read TeX. Alternatives include waiting until I compile this to PDF, or learning to read TeX. It isn't so difficult!

Consider the map $D:L(V,W) \To L(W^*, V^*)$ given by $D(T) = T^*$. Show $D$ is an isomorphism.

Consider the map $tr:M_{n \times n}(\Field) \To \Field$ and the map $\kappa: \Field \To M_{n \times n}(\Field)$ defined by $\kappa(a) = a I$. What is $tr(\kappa \compose tr)$?

Suppose $T$ is a $3 \times 3$ matrix, which has exactly two eigenvalues, both of which are integers, and is diagonalisable. Suppose further that $det(T) = 4, tr(T)=5$. What are the eigenvalues of $T$? (Do you need to know $T$ is diagonalisable to answer this?)

Say $T : V \To V$, and $V$ is three dimensional. Suppose $T$ has a fixed point other than $0$, and $det(T) = tr(T) = 6$. What are the eigenvalues of $T$?

For a matrix $A$, define $exp(A) = 1 + A + 1/2! A^2 + 1/3! A^3 + \cdots$. If $A$ is diagonalisable, prove $det(exp(A)) = e^{tr(A)}$. (This is actually true even when $A$ is not diagonalisable -- but I think this is harder to prove!)

We say a linear map $\varphi : M_{n \times n}(\Field) \To \Field$ is \emph{left-faithful} if $\varphi(xy) = 0$ for all $y$ implies $x = 0$, and \emph{right-faithful} if $\varphi(xy) = 0$ for all $x$ implies $y = 0$. Prove that right-faithfulness is actually the same thing as left-faithfulness!

Hints: \begin{enumerate} \item Define a map $\zeta : M_{n \times n}(\Field) \ To M_{n \times n}(\Field)^*$ by $\zeta(x)(y) = \varphi(xy)$. \item Show that if $\varphi$ is left-faithful, then $\zeta$ is injective. \item Think abot the dimensions of the domain and codomain of $\zeta$. What else can we say about $\zeta$? \item Show that if $\varphi(xy)=0$ for all $x$, then $y$ is killed by \emph{every} linear functional on $M_{n \times n}(\Field)$. What does this say about $y$? \end{enumerate}