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Exam 1 is \emph{Wednesday, November 22} in class.  It will primarily
cover Chapter 4 (sections 4.1 though 4.7) and Chapter 5 (sections 5.1
though 5.4).  Of cousre this material is based on the earlier material
and hence you should be comfortable with the earlier material.

Another good source of practice problems for the exam can be found in
the homework problems assigned (especially the uncollected ones, if
you didn't do them yet $\ldots$).  The quizes are also a good place to
look.

The exam will have some definitions, some true/false with reasons
problems, some computations problems and one or two short proof-like
problems. Very similar to the last exam.

Prof.\ Cholak (office: 223 CCMB) will have drop in office hours during
the day on Tuesday (on November 21 and 28) from 1 pm to 5 pm.  Prof
Wong (office: 234 CCMB) will have drop in office Tuesday (on November
21 and 28) from 8 pm to 10 pm.\\

\begin{problem} \emph{Possible definitions and explanatory material.}
  \begin{enumerate}
        \item If $\mathcal{B}$ is a basis for a vector space $V$, and
      $\vec{x}$ is a vector in $V$, what is the coordinate vector
      $[\vec{x}]_{\mathcal{B}}$? To which vector space does
      $[\vec{x}]_{\mathcal{B}}$ belong?
    \item If $\mathcal{B}$ is a basis for $\mathbb{R}^n$, what is
      meant by the \emph{change-of-coordinates matrix }
      $P_{\mathcal{B}}$? What does $P_{\mathcal{B}}$ enable one to
      compute?
    \item what is meant by a \emph{finite dimensional} vector space?
      by an \emph{infinite dimensional} vector space?
    \item If $A$ is an $m \times n$ matrix, define the \emph{row
        space} of $A$.  Define the rank of $A$ (note that by Theorem
      14 in \S4.6 there are several equivalent definitions of the rank
      of $A$. That theorem is pretty important.)
  \item You should know how to obtain ``change-of-basis'' matrices as
    in \S4.7. If $\mathcal{B}$ and $\mathcal{C}$ are bases for a
    vector space $V$, let $P$ denote the change of basis matrix. For a
    vector $\vec{v}$ in $V$, you should be able to state the
    relationship between $[\vec{v}]_\mathcal{B}$ with
    $[\vec{v}]_\mathcal{C}$ using the matrix $P$.
  \item If $A$ is a square ($m \times m$) matrix, what is meant by an
    eigenvector for $A$? an eigenvalue? the characteristic polynomial?
  \item You should be familiar with the following fact: if
    $\vec{v}_1,\dots,\vec{v}_n$ are eigenvectors for $A$ with
    \emph{distinct} eigenvalues $\lambda_1,\dots,\lambda_n$ (distinct
    means that if $i \not = j$ then $\lambda_i \not = \lambda_j$), then
    the vectors $\vec{v}_i$ are linearly independent.
  \item If $A$ and $B$ are square matrices of the same size, what does
    it mean to say that $A$ and $B$ are \emph{similar}? Be sure you
    know the difference between similarity of matrices (in this sense)
    and our older notion of (row) equivalent matrices.
  \item If $A$ is a square matrix, what does it mean to say that $A$
    is diagonalizable? There are several equivalent formulations. You should
    know how to obtain the diagonal matrix similar to $A$ (eigenvalues...)
    as well as an invertible matrix $P$ which defines the necessary
    similarity transformation. 
  \item You should remember that if an $m\times m$ matrix has $m$
    distinct eigenvalues it is diagonalizable. (why is that?)
    Moreover, you should have on hand examples of matrices which are
    \emph{not} diagonalizble.
  \item If $T: V \to W$ is a linear transformation and $\mathcal{B}$
    and $\mathcal{C}$ are bases for $V$ and $W$ respectively, what is
    meant by the matrix of $T$ relative to $\mathcal{B}$ and
    $\mathcal{C}$? If $M$ is this matrix, you should be able to use
    $M$ to relate $[\vec{v}]_\mathcal{B}$ and
    $[T(\vec{v})]_\mathcal{C}$ for any vector $\vec{v}$ in $V$.
     \end{enumerate}
\end{problem}

\newpage

\begin{problem}
  True/False: (as usual, if the statement is true give a reason,
  otherwise provide an example showing that it is false.)
  \begin{enumerate}
   \item Every set of 5 vectors in $\mathbb{R}^4$ is linearly dependent.
  \item There is a subspace $H$ of $\mathbb{R}^5$ with $\dim H = 6$.
  \item If $A$ is an $m \times m$ matrix where $m \ge 2$, and $A$ has
    only 1 eigenvalue, then $A$ is not diagonalizable.
  \item If the $m \times m$ matrix $A$ is similar to $B$, then $I_m + A$
    is similar to $I_m + B$.
  \item If $v$ and $w$ are eigenvectors for the $m \times m$ matrix $A$
    then $v + w$ is again an eigenvector for $A$.
  \item If $v$ is an eigenvector for the $m \times m$ matrix $A$,
    then $v$ is an eigenvector for $A^m$ for any integer $m \ge 0$.
  \item Let $A$ be a $5 \times 3$ matrix. If the rank of $A$ is 3, then
    $A$ defines a one-to-one linear transformation $\R^3 \to \R^5$.
  \item If $A$ is a diagonalizable $4 \times 4$ matrix with
    eigenvalues $\lambda=3$ and $\lambda=0$, then $A$ has rank 2.
  \end{enumerate}
\end{problem}

\begin{problem}
  Let $A = 
  \begin{bmatrix}
    1 & 0 & 1 & 2 \\
    0 & 1 & 3 & 0 \\
    0 & 0 & 0 & 1
  \end{bmatrix}.$
  \begin{subproblem}
    Find a basis for $\text{Nul}(A)$. What is the dimension of this subspace?
  \end{subproblem}
  \begin{subproblem}
    Find a basis for $\text{Col}(A)$. What is the dimension of this subspace?
  \end{subproblem}
  \begin{subproblem}
    What is the rank of $A$?
  \end{subproblem}
\end{problem}

\begin{problem}
  Let $M$ be a $8 \times 9$ matrix, and assume that the null space of $M$
  is 4 dimensional. How many pivots does $M$ have? What is the dimension
  of the row space of $M$?
\end{problem}

\begin{problem}
  Let $H$ be the column space of $A$ where $ = 
  \begin{bmatrix}
    1 & 2 & 0 & 1 \\
    0 & 1 & 3 & 1 \\
    0 & 0 & 1 & 0
  \end{bmatrix}$
  \begin{subproblem}
    Find a basis $\mathcal{B}$ for $H$.
  \end{subproblem}
  \begin{subproblem}
    For each of the 4 columns $\vec{a_i}$ ($1\le i \le 4$) of $A$,
    find $[\vec{a_i}]_{\mathcal{B}}$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Let $\mathcal{B} = \left \{ 
    \begin{bmatrix}
    1 \\ 1
    \end{bmatrix},
    \begin{bmatrix}
      1 \\ -2
    \end{bmatrix}
  \right \}$.
  \begin{subproblem}
    Show that $\mathcal{B}$ is a basis for $\mathbb{R}^2$
  \end{subproblem}
  \begin{subproblem}
    If $\vec{x}$ is a vector in $\mathbb{R}^2$ with 
    $[\vec{x}]_{\mathcal{B}} = 
    \begin{bmatrix}
      2 \\ 1
    \end{bmatrix}$, find $\vec{x}$.
  \end{subproblem}
  \begin{subproblem}
    If $\vec{x} = 
    \begin{bmatrix}
      2 \\ 1
    \end{bmatrix}$, find $[\vec{x}]_{\mathcal{B}}$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Let $\mathcal{E} = \{1,t,t^2,t^3\}$ be the standard basis for
  $\mathbf{P}_3$, the space of polynomials of degree $\le 3$.
  \begin{subproblem}
    Let $\mathbf{p}(t) = 1 + t^2 + t^3$, and $\mathbf{q}(t) = 1 + 2t +
    3t^3.$ Find the coordinate vectors $[\mathbf{p}(t)]_{\mathcal{E}}$
    and $[\mathbf{q}(t)]_{\mathcal{E}}$.
  \end{subproblem}
  \begin{subproblem}
    Explain how to use (a) to show that
    $\{\mathbf{p}(t),\mathbf{q}(t)\}$ form a linearly independent
    subset of $\mathbf{P}_3$.
  \end{subproblem}
\end{problem}


\begin{problem}
  Let $A,B,C$ be $m\times m$ matrices. Assume that $A$ and $B$ are
  similar, and that $B$ and $C$ are similar. Carefully prove that $A$
  and $C$ are similar.
\end{problem}

\begin{problem}
  Let $\vec{u}_i$ be eigenvectors for $\lambda_i$, for $i=0,1,2$.
  Assume that $\lambda_j \neq \lambda_k$.  Show that $\vec{u}_0,
  \vec{u}_1,\vec{u}_2$ are linearly independent.
\end{problem}

\newpage

\begin{problem}
  Let $A = 
  \begin{bmatrix}
    3 & 2 \\
    2 & 2
  \end{bmatrix}.$
  Decide if $A$ is diagonalizable. If it is, find invertible $P$ and
  diagonal $D$ so that $A = PDP^{-1}$.
\end{problem}

\begin{problem}
  Consider the vector space $V = \mathbf{M}_2$ of $2 \times 2$
  matrices.  Let
  \begin{equation*}
    \mathcal{B} = \left \{ 
  E_{1,1} = \begin{bmatrix}
    1 & 0 \\
    0 & 0
  \end{bmatrix},
  E_{1,2} = \begin{bmatrix}
    0 & 1 \\
    0 & 0
  \end{bmatrix},
    E_{2,1} = \begin{bmatrix}
    0 & 0 \\
    1 & 0
  \end{bmatrix},
  E_{2,2} = \begin{bmatrix}
    0 & 0 \\
    0 & 1
  \end{bmatrix}
\right\}.
\end{equation*}
\begin{subproblem}
  Show that $\mathcal{B}$ is a basis for $V$. (So what is $\dim V$?)
\end{subproblem}
\begin{subproblem}
  Let $A = 
  \begin{bmatrix}
    1 & 1 \\
    0 & 2
  \end{bmatrix}$.
  Let $T:V \to V$ be given by the rule
  $T(M) = AM - M$.
  Show that $T$ is a linear transformation.
\end{subproblem}
\begin{subproblem}
  Compute the matrix $N$ of $T$ relative to the basis $\mathcal{B}$.
  Note that $N$ is a $4 \times 4$ matrix, not a $2 \times 2$ matrix!!!
\end{subproblem}
\begin{subproblem}
  Find a basis of the nullspace of $N$, and find the corresponding
  ``vectors'' in $V$ which lie in the kernel of $T$.
\end{subproblem}
\end{problem}


\begin{problem}
  Let $0 \not = h$ be a constant, and consider the matrix
  \begin{equation*} 
    A = 
    \begin{bmatrix}
      1 & 1 & 1/h & 1 \\
      0 & 2 & h & 1 \\
      0 & 0 & 1 & 1 \\
      0 & 0 & 0 & 3 \\
    \end{bmatrix}.
  \end{equation*}
  \begin{subproblem}
    For what values of $h$ does the $\lambda=1$ eigenspace
    have dimension 2?
  \end{subproblem}

  \begin{subproblem}
    If the $\lambda=1$ eigenspace has dimension 2, is $A$ diagonalizable? Why?
  \end{subproblem}

  \begin{subproblem}
    Find a basis for the $\lambda=3$ eigenspace of $A$.
  \end{subproblem}
\end{problem}


\begin{problem}
  Consider the basis $\mathcal{B}$ of $\mathbb{R}^2$ whose vectors are
  $\vec{b_1} =
  \begin{bmatrix}
    1 \\
    2 + \sqrt{5}
  \end{bmatrix}$ and $\vec{b_2} = 
  \begin{bmatrix}
    1 \\
    2 - \sqrt{5} 
  \end{bmatrix}$.  Let 
  \begin{equation*}
A = 
  \begin{bmatrix}
    -2 & 1 \\
    1 & 2
  \end{bmatrix}.
\end{equation*} 
  \begin{subproblem}
    Find the $\mathcal{B}$ coordinate vectors
    $[A\vec{b_1}]_\mathcal{B}$ and     $[A\vec{b_2}]_\mathcal{B}$ .
  \end{subproblem}
   \begin{subproblem}
    Find the matrix of $A$ relative to the basis $\mathcal{B}$.
  \end{subproblem}
   \begin{subproblem}
    Let $\vec{v} = \vec{b_1} + 2 \vec{b_2} = 
    \begin{bmatrix}
      3 \\
      6 - \sqrt{5}
    \end{bmatrix}$.  Solve the equation $A\vec{x} = \vec{v}$ for $\vec{x} = c_1 \vec{b_1} + c_2 \vec{b_2}$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Let $A$ be a $5 \times 3$ matrix, and let
  \begin{equation*}
\vec{v_1} = 
  \begin{bmatrix}
    1 \\ 
    2 \\ 
    0 \\
    0 \\
    0 \\
  \end{bmatrix}, \quad 
  \vec{v_2} = \begin{bmatrix}
    0 \\ 
    1 \\ 
    3 \\
    0 \\
    0 \\
  \end{bmatrix}, \quad \vec{v_3} = \begin{bmatrix}
    0 \\ 
    0 \\ 
    1 \\
    4 \\
    0 \\
  \end{bmatrix}.
\end{equation*}
Suppose that the equations $A\vec{x} = \vec{v_i}$ have solutions for 
  $i = 1,2,3$. Prove that the nullspace of $A$
  is $\{0\}$. (Hint: what can be said about the rank of $A$?)
\end{problem}



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