\documentclass{amsart}

\title{Oral Final Exam Review}

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\maketitle

\section*{Theorems \& Results You Have to Know}
\begin{enumerate}
\item The 3 Isomorphism Theorems
\item The universal property of the quotient (explain how to construct the map from the quotient and why it gives a bijection between the relevant sets)
\item The universal property of the tensor product (explain how to construct maps from the tensor product and why it gives an identification with bilinear maps)
\item Riesz representation theorem for a non-degenerate bilinear form
\item The theorem about cyclic subspaces for nilpotent spaces (the one that requires the top non-zero powers of a linear operator on the set be linearly independent).
\item Choice of a basis gives an isomorphism with a free vector space. Choice of an orthonormal basis gives an isomorphism with $\mathbb R^n$ or $\mathbb C^n$ with the dot product.
\item The dual of a short exact sequence is short exact
\end{enumerate}

\section*{Things You Must Be Able to Compute}
\begin{enumerate}
\item Jordan form and Jordan basis
\item Gram-Schmidt orthogonalization
\item The value of the exterior powers of a linear operator on various elements
\end{enumerate}

That's it!

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