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\begin{document}



\title{Algebra Review Problems}

\author{}

\date{}







\maketitle


{\bf The problems below are very basic ones. It is served as a ``self-service
placement test'': you would need to work on the subject more if you did not
feel comfortable with them.}



\section{Algebra}
\begin{enumerate}
\item Let $\Q/\Z$ be the set
    \[
    \left\{ a \in \Q \mid 0 \leq a < 1 \right\},
    \]
    and define
    \[
    a\oplus b = \text{ the fraction part of $a+b$}.
    \]
    Show that $(\Q/\Z, \oplus)$ is an abelian group.

\item Find all abelian groups of order 16 and 99.

\item Show that
    \[
    \Q(\sqrt[3]{2}) = \{ a+ b\sqrt[3]{2}+ c\sqrt[3]{4} \mid
    a, b, c \in \Q\},
    \]
    with the usual addition and multiplication is a field.

\item Explain why $\Z/n\Z$, with the usual addition and multiplication mod
    $n$ is not a field if $n$ is not a prime.

\item Let $p$ be a prime. Show that $(x+y)^p = x^p + y^p$ if $x, y \in
    \Z/p\Z$. Explain why this is not the case if $p$ is not a prime.
\end{enumerate}



\section{Vector spaces}
\begin{enumerate}
\item (Cartan matrix) Find the determinant of the matrix
    \[
    \begin{pmatrix} 2 & -1  & 0 & \cdots & 0 \\
    -1 & 2 & -1 & \ddots & \vdots\\
    0 & \ddots & \ddots & \ddots & 0 \\
    \vdots & \ddots & -1 & 2 & -1 \\
    0 & \cdots & 0 & -2 & 2 \\
    \end{pmatrix}
    \]

\item Let $V$ be the space of polynomials of degree at most $n$. Define a
    linear operator on $V$ by
        \[
        D: V \to V, \quad D(p(x)) = p'(x) + 2 p''(x).
        \]
    \begin{enumerate}
    \item What are the kernel and image of $D$? What are their
        dimensions?

    \item Find two different bases of $V$ and write down the matrices
        representing $D$ under these two bases. Can you make the relation
        between these two matrices explicit?

    \item What is the characteristic polynomial of $D$? What are the
        trace and determinant of $D$?

    \item Find a basis so that the matrix representing $D$ is in Jordan
        canonical form.
    \end{enumerate}

\item Suppose that $A = \begin{pmatrix} 0 & 2 & -3 \\ -1 & 3 & -3 \\1
    & -2 &a \end{pmatrix}$ is similar to $B = \begin{pmatrix} 1 & -2 & 0 \\
    0 & b & 0 \\ 0 & 3 & 1\end{pmatrix}$.
    \begin{enumerate}
    \item Find $a$ and $b$.

    \item Find an invertible matrix $P$ so that $P^{-1} A P$ is in Jordan
        canonical form.
    \end{enumerate}

\item Show that a square matrix is similar to its transpose.
\end{enumerate}






\section{Euclidean spaces}
\begin{enumerate}
\item Let $V$ be a finite dimensional Euclidean space over $\mathbb{R}$.
    Let $T: V \to V$ be a linear transform and $T^*$ be its adjoint. Assume
    that $TT^* = T^* T$. Show that if $\alpha, \beta \in V$ with
    $T(\alpha)+ \alpha = T(\beta) = 0$, then $\alpha \perp \beta$.

\item Show that
    \[
    (A, B) \mapsto \Tr \tp{A} B
    \]
    defines an inner product on the space of $n \times n$ real matrices.

\item Find a QR-decomposition of
    \[
    \begin{pmatrix} 12 & -51 & 4 \\
    6 & 167 & -68\\
    -4 & 24 & -41
    \end{pmatrix}, \quad
    \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix}.
    \]

\item Find a decomposition $A = S D S^{-1}$ for
    \[
    A = \begin{pmatrix} -1 & 1 & 2 \\ 1 & -1 & -2 \\ 2 & -2 & -4 \end{pmatrix},
    \quad \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{pmatrix},
    \]
where $S$ is an orthogonal matrix and $D$ is diagonal.


\item Find a singular value decomposition of
    \[
    \begin{pmatrix}
    1 & 0 & 0 & 0 & 2\\
    0 & 0 & 3 & 0 & 0\\
    0 & 0 & 0 & 0 & 0\\
    0 & 2 & 0 & 0 & 0
    \end{pmatrix}, \quad
    \begin{pmatrix}
    2 & 4 \\ 1 & 3\\ 0 & 0 \\ 0 & 0 \end{pmatrix},
    \quad
    \begin{pmatrix}
    3 & 2 & 2 \\ 2 & 3 & -2 \end{pmatrix}.
    \]
\end{enumerate}



\section{Multilinear algebra}

\begin{enumerate}
\item Find the signature of
    \[
    q(x_1, x_2, x_3) = x_1^2 + 2x_2^2 + 5x_3^2 + 4 x_1 x_2 + 7 x_2 x_3 -8 x_1 x_3.
    \]

\item Find the signature of
    \[
    q(x_1, \cdots, x_n) = \sum_{i = 1}^{[(n-1)/2]} x_i x_{n+1-i}.
    \]

\item Prove the universal property of tensor products.

\item Work out the definition of tensor product of finitely many vector
    spaces in detail.
\end{enumerate}


\section{Slightly harder problems}

\begin{enumerate}
\item (Rational canonical form) Find the minimal polynomial of
    \[
    \begin{pmatrix}
    0 &&&& a_0\\
    1 & 0 &&& \vdots\\
    & 1 & \ddots && \vdots\\
    && \ddots & 0 & a_{n-2} \\
    &&& 1 & a_{n-1} \end{pmatrix}
    \]
    Here $a_0, \cdots, a_{n-1}$ are complex numbers.

\item (Cayley--Hamilton Theorem) Let $A$ be an $n \times n$ matrix and $f$
    be its characteristic polynomial. Then $f(A) = 0$.

\item (Fitting's Lemma) Let $f: V \to V$ be a linear transform. Then there
    are invariant subspaces $V_1$ and $V_2$ of $V$ so that $V = V_1 \oplus
    V_2$ and $f|_{V_1}$ is invertible and $f|_{V_2}$ is nilpotent.

\item Let $A$ be an $n \times n$ complex matrix. Then there is a $B$ so
    that $A = B^2$.

\item Let $A$ and $B$ be $n \times n$ matrices and assume that $AB = BA$.
    So that there is a nonzero $v \in \C^n$ which is an eigenvector for
    both $A$ and $B$.

\item Let $A$ be a (real) square matrix. Show that $A\tp{A} = \tp{A} A$ if
    and only if there is a polynomial $f$ so that $\tp{A} =f(A)$.

\item Let $A$ and $B$ be $n \times n$ real matrices. Assume that $A\tp{A} =
    \tp{A} A$ and $AB = BA$. Show that $A \tp{B} = \tp{B} A$.

\item Let $A$ be an $n \times n$ real symmetric matrix. Find the maximum
    and minimum of
    \[
    f(x) = \frac{\langle Ax, x\rangle}{\langle x, x\rangle},
    \quad x \in \R^n, \ x \not=0.
    \]

\item Let $A$ and $B$ be $n \times n$ positive definite symmetric matrices.
    Show that
    \[
    \det(A+B)^{\frac{1}{n}} \geq \det(A)^{\frac{1}{n}} + \det(B)^{\frac{1}{n}}.
    \]
\end{enumerate}








\end{document}
