\documentclass[12pt]{article}
\usepackage{amstex}
\input{ExamMacros.tex}

\prof{}
\class{Math 221}
\quizdate{}
%\examname{Practice Problems}

\begin{document}
\centerline{\large Some Practice Problems for Exam 3 -- Math 221}
\bigskip\bigskip
%\pagestyle{empty}

The exam will cover everything since the last exam:
5.5,5.6,6.1,6.2,6.3,6.5.  We did NOT cover the QR-dcomposition in
6.3. There will be somewhere between 20 and 30 points on
multi-guess problems on this exam.  Since this has put together
quickly the answers and numbers used need not be nice.

\vspace{1cm}

\begin{problem}
  Here are some problems from the text you should be able to
  solve:  5.5: 5,6,8,9,10,12; 5.6: 4,5,6,14; 6.1: 9,16; 6.2:
  8,9,12,16,18-21 (good multi-guess type problems); 6.3:
  13,19,21; 6.5: 10,22.
\end{problem}

\begin{problem}
  Consider the following bases of $\mathbb{R}^2$:  \\
  $\vec{u}_1 = 
  \begin{bmatrix}
    1 \\ -2
  \end{bmatrix}$,
  $\vec{u}_2 =
  \begin{bmatrix}
    1 \\ 1
  \end{bmatrix}$
  and $\vec{v}_1 = 
  \begin{bmatrix}
    \sqrt{2}/2 \\ \sqrt{2}/2
  \end{bmatrix}$,
  $\vec{v}_2 = \begin{bmatrix}
    \sqrt{2}/2 \\ -\sqrt{2}/2
  \end{bmatrix}$.
  

  \begin{subproblem}
    Express $\vec{u}_1 = a \vec{v}_1 + b \vec{v}_2$. (HINT: is $\vec{v}_1, \vec{v}_2$
    a nice basis?)
  \end{subproblem}

  \begin{subproblem}
    Express $\vec{u}_2 = c \vec{v}_1 + d \vec{v}_2$.
  \end{subproblem}

  \begin{subproblem}
    Let $A = 
    \begin{bmatrix}
      a & c \\
      b & d \\
    \end{bmatrix}$.
    Let $\vec{x} = 3 \vec{v}_1 + 2 \vec{v}_2$. Use the matrix $A$ to express
    $\vec{x}$ as a linear combination of $\vec{u}_1$ and $\vec{u}_2$.
  \end{subproblem}

\end{problem}

\begin{problem}
  Consider the matrix $A = 
  \begin{bmatrix}
    a_1 & a_2 & a_3 \\
    b_1 & b_2 & b_3 
  \end{bmatrix}$. 
  Assume that $A$ is row equivilent to the matrix $R =
  \begin{bmatrix}
    1 & 2 & 0 \\
    0 & 0 & -2 \\
  \end{bmatrix}$

  \begin{subproblem}
    Find a basis for the column space of $A$.
  \end{subproblem}
  

  \begin{subproblem}
    Find a basis for the row space of $A$.
  \end{subproblem}

  \begin{subproblem}
    Is the vector $\begin{bmatrix} -2 \\ 1 \\ 0
    \end{bmatrix}$ 
    in the null space of $A$? Answer ``yes'',
    ``no'', or ``can't tell''.  Explain your answer.
  \end{subproblem}


 \begin{subproblem}
    Is the vector $
    \begin{bmatrix}
      2 \\
      \sqrt{5}
    \end{bmatrix}$ 
    in the column space of $A$? Answer ``yes'',
    ``no'', or ``can't tell''.  Explain your answer.
  \end{subproblem}

\end{problem}

\begin{problem}
  Let $S$ be a basis for an inner product space $V$.  Let
  $\vec{u}$ and $\vec{v}$ be vectors in $V$ such that
  $(\vec{u})_S = (1,2,3,4)$ and $(\vec{v})_S = (-4,3,-2,1)$.
  Answer the following: What is the dimension of $V$? The norm of
  $\vec{u}$? of $\vec{u}$? Are $\vec{u}$ and $\vec{v}$
  orthogonal?  Compute the following: the norm of
  $4\vec{u}-2\vec{v}$, $<2\vec{u},-3\vec{v}-\vec{u}>$.  Find the
  coordinates of a vector which is orthogonal to both $\vec{u}$
  and $\vec{v}$.
\end{problem}

\begin{problem}
  Using Gram-Schmidt process to transform the following basis to
  an orthonormal basis.
  
  ${\bf u_1}=(0,0,1), {\bf u_2}=(1,1,2), {\bf u_3}=(1,2,2)$

\end{problem}

\begin{problem}
  Let $M = 
  \begin{bmatrix}
    1 & -1 & 3 & 0 \\
    2 & -1 & 3 & 1 \\
    1 & 0 & 0  & 1 \\
  \end{bmatrix}$.

  \begin{subproblem}
    Find a basis for the null space of $M$
  \end{subproblem}

  \begin{subproblem}
    Show that the vector $\vec{y} = 
    \begin{bmatrix}
      3\\ 5\\  2
    \end{bmatrix}$
    is in the column space of $M$.
  \end{subproblem}

  \begin{subproblem}
    Is the equation $M \vec{x} = \vec{y}$ consistent?  If so, find the
    general solution.
  \end{subproblem}

  \begin{subproblem}
    Find the rank of $M$.  Find the nullity of $M$.
  \end{subproblem}
\end{problem}


\begin{problem}
  Let $ A = \left(\begin{array}{cccc}
      1   &  1   & 3 & -1\\
      4  &  5 &   6  & -2\\
      5 & 6 & 9 & -3
\end{array}\right)
$ 

(a). Find a basis for the null space.


(b). Find a basis for the row space.
\end{problem}

\begin{problem}
  Find the rank and nullity of the given matrix.
  
  (a) $ \left(\begin{array}{cccc}
      1   &  1   & 0 & 0\\
      1  &  2 &   1  & -2\\
      2 & 3 & 1 & -2
    \end{array}\right)$
  
  (b) $ \left(\begin{array}{ccc}
      1 &  0  & -1\\
      2 &  0  & -2\\
      6 & 0 & -6
 \end{array}\right)
$
\end{problem}

\begin{problem}
  Multiple choice questions.
  
  (i). Suppose that $A$ is a $3\times 4$ matrix with Rank$(A)=2$
  and Rank$(A|{\bf b})=2$. Then
  
  (A). the system $A{\bf x} ={\bf b}$ is consistent,
  
  (B). the system $A{\bf x} ={\bf b}$ is not consistent,
  
  (C).  the consistency of the system $A{\bf x} ={\bf b}$ depends
  on ${\bf b}$. \\
  
  (ii).  Suppose that $A$ is a $4\times 6$ matrix, the system
  $A{\bf x} ={\bf b}$ is consistent, and Rank$(A)=3$. Then the
  number of parameters in the general solution is
  
  (A). 1, (B). 2, (C). 3, (D). 4.
  
  (iii).  If $A$ is a matrix with more rows than columns, then
  
  (A). Dim(row space)$<$Dim(column space),
  
  (B). Dim(row space)$>$Dim(column space),
  
  (C). Dim(row space)$=$Dim(column space).
  
  (iv). If $A=({\bf c_1, c_2, c_3, c_4}) $ is a $5 \times4$
  matrix and $\bf b = {\bf c_1} - 2{\bf c_2} +3{\bf c_3} -4{\bf
    c_4} $, then
 
  (A). the system $A{\bf x} ={\bf b}$ is consistant,
 
  (B). the system $A{\bf x} ={\bf b}$ is not consistant,
 
  (C).  the consistancy of the system $A{\bf x} ={\bf b}$ depends
  on ${\bf b}$.

\end{problem}


\begin{problem}
  Determine which of the following are inner product.  Circle
  your answer.
  
  (i). the space $R^3$, $<{\bf u, v}> = u_2v_2 + u_3v_3$, (here
  ${\bf u}=(u_1, u_2,u_3)$ and ${\bf v}=(v_1, v_2, v_3)$), ( Yes,
  No).
  
  (ii). the space $R^3$, $<{\bf u, v}> = u_1v_1+2u_2v_2 +
  3u_3v_3$, (here ${\bf u}=(u_1, u_2,u_3)$ and ${\bf v}=(v_1,
  v_2, v_3)$), ( Yes, No).
  
  (iii). the space $R^3$, $<{\bf u, v}> = u_1^2 v_1^2+ u_2v_2 +
  u_3^2v_3^2$, (here ${\bf u}=(u_1, u_2,u_3)$ and ${\bf v}=(v_1,
  v_2, v_3)$), ( Yes, No).
  
  (iv). the space $R^3$, $<{\bf u, v}> = u_1v_1 + u_2v_2 -
  u_3v_3$, (here ${\bf u}=(u_1, u_2,u_3)$ and ${\bf v}=(v_1, v_2,
  v_3)$), ( Yes, No).

\end{problem}
 

\begin{problem}
  Suppose that $<{\bf u, v}> =1, \|{\bf u}\| =2, \|{\bf v}\|=3$
  Find $\|{\bf u+ v}\|$.
\end{problem}

\begin{problem}
  Let $W$ be the line in $R^2$ with equation $y = 5 x$. Find the
  equation for $W^\perp$.
\end{problem}



\end{document}


