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Exam 1 is \emph{Friday, October 13.} It will cover Chapter 1 (sections
1.1 - 1.8), Chapter 2 (sections 2.1 - 2.3), Chapter 3 (sections 3.1 -
3.3) and the first third sections of Chapter 4.  A 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...).  The quizes are also 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.

Prof.\ Cholak (office: 223 CCMB) will have drop in office hours during
the day on Thursday from 9 am to 11 am and 1 pm to 5 pm.  Prof Wong
(office: 234 CCMB) will have drop in office Thrusday from 8 pm to 11
pm. \\


\begin{problem}
  Make sure you know the definitions of the terms used in the class.
  Here are some that you might keep in mind (this list might not
  include everything that you need to know!). Note that it is often
  very useful to have in mind some examples; as this is a useful way
  to help you remember terminology, you might think about what
  examples seem most natural to you.
  \begin{itemize}
  \item What does it mean for a linear system to be consistent? 
  \item What is an elementary row operation?
  \item What are row echelon and reduced row echelon matrices?
  \item What is a pivot position in a matrix? pivot column?
  \item What is a free variable? general solution?
  \item What is a linear combination? What is the \emph{span} of a set
    of vectors? What does it mean for some vectors to \emph{span}
    $\mathbb{R}^n$?
  \item If $\vec{v}$ and $\vec{w}$ are vectors in $\mathbb{R}^n$, what
    is $\vec{v} + \vec{w}$? What is $c\vec{v}$ for a scalar $c$?
  \item What is a homogeneous system?
  \item What is linearly independence?
  \item What is a linear tranformation? If $T$ is a linear
    transformation, what is meant by the \emph{domain} of $T$? the
    \emph{range} of $T$? the \emph{codomain} of $T$? What does it mean to say
    that $T$ is one-to-one? that $T$ is onto?
  \item What is the standard matrix of a linear transformation?
  \item If $A$ and $B$ are matrices, when can (and how does...) one
    define $A+B$? $A\cdot B$? $rA$ where $r$ is a scalar? What is $A^T$?
  \item What does it mean to say that a matrix $A$ is invertible?
  \item What is an elementary matrix?
  \item If $A$ is an $n\times n$ matrix and $1 \le i,j \le n$, what is
    meant by the notation $A_{i,j}$? $C_{i,j}$? what is a co-factor?
    \item Give the formula for $\det A$ using cofactor expansion along
      the $i$th row, where $1 \le i \le n$ is fixed. Then give the
      definition using cofactor expansion along the $j$th column,
      where $1 \le j \le n$ is fixed.
    \item You should review the definition of a vector space. I'm not
      likely to ask you to list all 10 axioms, but you should
      certainly be familiar with them.\newpage
    \item Define a \emph{subspace} of a vector space $V$.
    \item Let $A$ be an $m \times n$ matrix. Define the null space
      $\text{Nul}(A)$. What vector space is the null space a subspace
      of? Define the column space $\text{Col}(A)$. What vector space is the
      column space a subspace of? 
    \item define the \emph{kernel} of a linear transformation $T:V \to
      W$, where $V$ and $W$ are vector spaces.
    \item define a \emph{basis} for a vector space $V$.
    \item what is the \emph{standard basis} for $\mathbb{R}^n$? for
      $\mathbb{P}_n$?
\end{itemize}
\end{problem}

\begin{problem}
  Much of the first part of this course involves rephrasing problems
  in various ways. You should know how to translate! For example:
  \begin{itemize}
  \item  Given a system of linear equations, you should know how to
    produce the corresponding \emph{vector equation} and \emph{matrix
      equation} (also various permuations of this: e.g. instead
    \emph{start} with a matrix equation...)
  \item Be able to explain why a matrix equation $A\vec{x}=\vec{b}$ is
    consistent $\iff$ $\vec{b}$ is in the span of the columns of $A$
    $\iff$ the last column of the corresponding augemented matrix is not
    a pivot column.
  \item Explain why the vectors $\vec{v}_1,\dots,\vec{v}_m$ are linearly
    independent if and only if the homogeneous matrix equation $A\vec{x}
    = \vec{0}$ has only the trivial solution, where $A = [\vec{v}_1
    \dots \vec{v}_m]$ is the matrix whose columns are the $\vec{v}_i$.
  \item Let $T$ be a linear transformation $T:\mathbb{R}^n \to
    \mathbb{R}^m$, and let $A$ be its standard matrix. (What size matrix
    is $A$?) Explain why $T$ is one-to-one if and only if the columns of
    $A$ are linearly independent. Explain why $T$ is onto if and only if
    the columns of $A$ span $\mathbb{R}^m$.\end{itemize}
\end{problem}
\begin{problem} True/False: (as usual, if the statement is true give a reason,
  otherwise provide an example showing that it is false.) Another good
  place to look is at the True/False in the book.
 
  \begin{subproblem} Let $\vec{u},\vec{v},\vec{w}$ be vectors in the vector space
    $V$, and let $H = \text{Span}(\vec{u},\vec{v},\vec{w})$.  Suppose
    that $\vec{u} + 2 \vec{v} + 3\vec{w} = \vec{0}$. Then
    $\{\vec{u},\vec{v}\}$ is a basis for $H$.
  \end{subproblem} \begin{subproblem} Let $A$ and $B$ be $m \times m$ matrices. Then $\det(A+B) =
    \det(A) + \det(B)$.
  \end{subproblem} \begin{subproblem} The subset $\left \{ 
      \begin{bmatrix}
        1 \\ a \\ a + b \end{bmatrix}:a, b \text{ real numbers} \right
    \}$ of $\mathbb{R}^3$ is a \emph{subspace}.
  \end{subproblem} \begin{subproblem} The set $\{\cos(\theta),\sin(\theta)\}$ form a linearly independent
    subset of the vector space of all continuous functions from
    $\mathbb{R}$ to $\mathbb{R}$.
  \end{subproblem} \begin{subproblem} 
    Suppose $\mathcal{S} = \{ \vec{v_1},\dots,\vec{v_n}\}$ spans the
    vector space $V$. Then some subset of $\mathcal{S}$ is a basis for
    $V$.
  \end{subproblem} \begin{subproblem} 
    Let $A$ be an $m \times n$ matrix. If $\text{Nul}(A) =
    \{\vec{0}\}$, then the columns of $A$ form a basis for the column
    space $\text{Col}(A)$.
  \end{subproblem}
\end{problem}

\begin{problem}
  Consider the equations
  \begin{align*}
    x - y +  z - w & = 0 \\
             z - 2w & = 0 
  \end{align*}
  \begin{subproblem}
    Describe the solution set $S$ as the span of a collection of
    vectors in $\mathbb{R}^4$.
  \end{subproblem}
  \begin{subproblem}
    Let $A = 
    \begin{bmatrix}
      1 & -1 & 1 & -1 \\
      0 & 0 & 1 & -2
    \end{bmatrix}$, and let $\vec{b_1} = 
    \begin{bmatrix}
      4 \\ 2
    \end{bmatrix}$, $\vec{b_2} = 
    \begin{bmatrix}
      17 \\ 2
    \end{bmatrix}$. Give  general solutions to the
    equations $A\vec{x} = \vec{b_1}$ and $A\vec{x} = \vec{b_2}$.  Use
    your answers to describe \emph{all} possible matrices $C$ with the
    property that $AC =
    \begin{bmatrix}
      17 & 4 \\
      2 & 2 
    \end{bmatrix}$.
  \end{subproblem}
\end{problem}

\begin{problem} 
  Give examples as directed; be sure to explain why your example does
  what you claim. On the exam, you \emph{will not} get full credit if
  you don't explain your examples!
  \begin{subproblem}
    Give an example of a linear transformation $T:\mathbb{R}^2 \to
    \mathbb{R}^3$ which is one-to-one but not onto.
  \end{subproblem}
  \begin{subproblem}
    Give an example of a linear transformation $T:\mathbb{R}^3 \to
    \mathbb{R}^2$ which is onto but not one-to-one.
  \end{subproblem} \newpage
  \begin{subproblem}
    Give an example of a linearly dependent set
    $\{\vec{v}_1,\vec{v}_2,\vec{v}_3,\vec{v}_4\}$ in $\mathbb{R}^4$
    such that each of the subsets
    $\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$,
    $\{\vec{v}_1,\vec{v}_2,\vec{v}_4\}$,
    $\{\vec{v}_1,\vec{v}_3,\vec{v}_4\}$, and
    $\{\vec{v}_2,\vec{v}_3,\vec{v}_4\}$ are linearly independent.
  \end{subproblem}
  \begin{subproblem}
    Give an example of a plane in $\mathbb{R}^3$ that contains neither
    the vector $\vec{e_1}=
    \begin{bmatrix}
      1 & 0 & 0
    \end{bmatrix}^T$ nor the vector $\vec{e_2}=
    \begin{bmatrix}
      0 & 1 & 0
    \end{bmatrix}^T$.
  \end{subproblem}
\end{problem}

\begin{problem}
  True/False questions. If the statement is true, give a reason. If it
  is false, provide an example which shows that it is false.
  \begin{subproblem}
    Suppose that $A$ and $B$ are $n \times n$ matrices, and that
    $AB=\vec{0}$.  Then $A$ is singular (i.e. $A$ is not invertible).
  \end{subproblem}

  \begin{subproblem}
    If $A$ is an invertible $n\times n$ matrix, then $A + I_n$ is also
    invertible.
  \end{subproblem}

  \begin{subproblem}
    If $A$ is an $m \times n$ matrix whose columns do not span
    $\mathbb{R}^n$, then the equation $A\vec{x} = \vec{b}$ is
    inconsistent for every $\vec{b}$ in $\mathbb{R}^n$.
  \end{subproblem}

  %\begin{subproblem}
%    Let $A$ be an $m \times n$ matrix. Then $B=AA^T$ is a square
%    matrix satisfying $B^T = -B$.
%  \end{subproblem}

  \begin{subproblem}
    Let $A$ be a $20 \times 30$ matrix. Suppose that $A\vec{x} =
    \vec{e_{19}}$ is a consistent equation. ($\vec{e_i}$ denotes the
    column whose $i$th entry is 1, and whose other entries are 0.)
    Then $A\vec{x} = \vec{e_{19}}$ has infinitely many solutions.
  \end{subproblem}

  \begin{subproblem}
    A $2 \times 10$ matrix always has at least 2 pivots.
  \end{subproblem}
\end{problem}

\begin{problem} Let $I=I_2$ be the $2 \times 2$ identity matrix.
  \begin{subproblem}
    Let $K$ be a $2 \times 2$ matrix satisfying $K^2 = -I$. Show that
    $K$ is invertible.
  \end{subproblem}
  \begin{subproblem}
    Let $J = \begin{bmatrix}
      0 & -1 \\
      1 & 0 
    \end{bmatrix}$. Show that $J^2 = -I$.
  \end{subproblem}
  \begin{subproblem}
    Let $P$ be an invertible $2 \times 2$ matrix.  Show that $K =
    PJP^{-1}$ satisfies $K^2 = -I$.
  \end{subproblem}
  \begin{subproblem}
    Show that there are infinitely many $2 \times 2$ matrices $K$ such
    that $K^2 = -I$. (use part (c)!)
  \end{subproblem}
\end{problem}

\begin{problem} 
  Find the inverses of the following matrices.
  \begin{subproblem} 
    $A = \begin{bmatrix}
      2 & 0 & 0 & 0 \\
      0 & 3 & 0 & 0 \\
      0 & 0 & -1 & 0 \\
      0 & 0 & 0 & \frac{1}{4}
    \end{bmatrix} \cdot
    \begin{bmatrix}
      0 & 1 & 0 & 0 \\
      1 & 0 & 0 & 0 \\
      0 & 0 & 0 & 1 \\
      0 & 0 & 1 & 0 
    \end{bmatrix}$    
  \end{subproblem}

\begin{subproblem} 
    $A = \begin{bmatrix}
      1 & -3 & -a \\
      0 & 3 & -6 \\
      0 & 0 & 1  
    \end{bmatrix}$
  \end{subproblem}
  \begin{subproblem}
    A=$\begin{bmatrix}
      \cos(\theta) & \sin(\theta) \\
      - \sin(\theta) & \cos(\theta)
    \end{bmatrix}$, \hspace{1cm} $0 \le \theta < 2\pi$.
  \end{subproblem}
\end{problem}

\begin{problem} Consider the  matrix
    $A=
    \begin{bmatrix}
      -1 & -2 \\
      1 & 0 \\
      0 & 1 
    \end{bmatrix}$

    \begin{subproblem}
      Does the equation $A \vec x = \vec b$ have a solution when
      $\vec b = 
      \begin{bmatrix}
        1 \\ 2 \\ 2
      \end{bmatrix}$? 
      
    \end{subproblem}
    \begin{subproblem}
      Does the equation $A \vec x = \vec b$ have a solution when
      $\vec b = 
      \begin{bmatrix}
        -4 \\ -2 \\ 3
      \end{bmatrix}$? 

    \end{subproblem}
    
    
\end{problem}


\newpage
\begin{problem}
  Compute the following determinants:
  \begin{itemize}
  \item $\det(A)$ where $A = 
    \begin{bmatrix}
      a_1 & 0 & 0 \\
      2 & a_2 & 0 \\
      1 & 2 & a_3 
    \end{bmatrix}$
  \item $\det 
    \begin{bmatrix}
      1 & 0 & 2 \\
      1 & 1 & 3 \\
      1 & 0 & 8
    \end{bmatrix}.$
\end{itemize}
\end{problem}
\begin{problem}
  Find an elementary $3 \times 3$ matrix $E$ so that $\det(E^2) =
  \sqrt{3}$.
\end{problem}

\begin{problem}
  Let $A = 
  \begin{bmatrix}
    0 & 1 & 0 \\
    1 & 0 & 1 \\
    0 & 1 & 0 
  \end{bmatrix}$.  For what scalars $c$ is the matrix
  $cI_3 - A$ an invertible matrix?
\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}
\end{problem}

\begin{problem}
  Find a matrix $A$ so that its column space $\text{Col}(A)$ is as
  follows:
  \begin{equation*}
    \text{Col}(A) = \left \{ 
      \begin{bmatrix}
      a + b + c \\
      2a - c \\
      3b + 4c
      \end{bmatrix}:a,b,c \text{ real numbers}
    \right \}
  \end{equation*}
\end{problem}

\begin{problem}
  Let $V$ be a vector space, and let $\vec{u}$ be a non-zero vector in
  $V$. If $c\vec{u}=\vec{0}$, prove that $c =0$.
\end{problem}

\begin{problem}
  Consider the function
  \begin{equation*}
    T:\mathbf{P}_2 \to \mathbb{R}^2
  \end{equation*}
  given by the rule $T(\mathbf{p}(t)) = 
  \begin{bmatrix}
    \mathbf{p}(0) \\
    \mathbf{p}(1) 
  \end{bmatrix}$
  \begin{subproblem}
    Show that $T$ is a linear transformation.
  \end{subproblem}
  \begin{subproblem}
    Find a basis for the kernel of $T$. (what vector space is the kernel
    a subspace of ??)
  \end{subproblem}
  \begin{subproblem}
    Show that $
    \begin{bmatrix}
      1 \\ 0
    \end{bmatrix}$ and 
    $\begin{bmatrix}
      0 \\ 1
    \end{bmatrix}$ are in the range of $T$.  Using this, describe the range 
    of $T$.
  \end{subproblem}
\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}$.
    Find a basis $\mathcal{B}$ for $H$.
  \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}
  
\end{problem}


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