%% This document created by Scientific Word (R) Version 3.0



\documentclass{article}

%\usepackage{sw20bams}

\usepackage{latexsym}

\usepackage{amsmath}

\usepackage{amsfonts}

\usepackage{amssymb}

\usepackage{graphicx}

%TCIDATA{OutputFilter=latex2.dll}

%TCIDATA{CSTFile=LaTeX article (bright).cst}

%TCIDATA{Created=Wed Sep 13 17:03:24 2000}

%TCIDATA{LastRevised=Mon Feb 05 11:58:20 2001}

%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}

%TCIDATA{<META NAME="DocumentShell" CONTENT="Journal Articles\Similar to Bulletin of the American Mathematical Society">}

\newtheorem{theorem}{Theorem}

\newtheorem{acknowledgement}[theorem]{Acknowledgement}

\newtheorem{algorithm}[theorem]{Algorithm}

\newtheorem{axiom}[theorem]{Axiom}

\newtheorem{case}[theorem]{Case}

\newtheorem{claim}[theorem]{Claim}

\newtheorem{conclusion}[theorem]{Conclusion}

\newtheorem{condition}[theorem]{Condition}

\newtheorem{conjecture}[theorem]{Conjecture}

\newtheorem{corollary}[theorem]{Corollary}

\newtheorem{criterion}[theorem]{Criterion}

\newtheorem{definition}[theorem]{Definition}

\newtheorem{example}[theorem]{Example}

\newtheorem{exercise}[theorem]{Exercise}

\newtheorem{lemma}[theorem]{Lemma}

\newtheorem{notation}[theorem]{Notation}

\newtheorem{problem}[theorem]{Problem}

\newtheorem{proposition}[theorem]{Proposition}

\newtheorem{remark}[theorem]{Remark}

\newtheorem{solution}[theorem]{Solution}

\newtheorem{summary}[theorem]{Summary}







\begin{document}





\author{B. Hall and R. Smarandache}

\title{Math 228, Test 2}

\date{October 24, 2000}

\maketitle







\textbf{Instructions}: You have one hour for the exam. There are 11 problems,

worth 9 points each, with one free point, for a total of 100 points possible.

For multiple-choice problems, please mark your answer clearly. For all other

problems, please \textit{show your work completely}. Partial credit will be

given for all non-multiple-choice problems.



Calculators, notes, and books are prohibited.



This exam is bound by the provisions of the Notre Dame Honor Code.\bigskip



\textbf{Name}: \bigskip



\begin{enumerate}

\item  Do the vectors $v_{1}=\left(  1,2,0\right)  $, $v_{2}=(0,1,2)$ and

$v_{3}=\left(  1,1,1\right)  $ constitute a basis for $\mathbb{R}^{3}$? Show

your work and explain clearly why or why not.\pagebreak 



\item  Let $u$ and $v$ be two vectors in $\mathbb{R}^{n}$ with $\left\|

u\right\|  =5$ and $\left\|  v\right\|  =6.$ What is the largest possible

value for $\left|  u\cdot v\right|  $? What is the largest possible value for

$\left\|  u+v\right\|  $? Explain your reasoning.\vspace{3.5in}



\item  Suppose $S=\left\{  v_{1},v_{2},v_{3}\right\}  $ is a linearly

\textit{dependent} set of vectors in $\mathbb{R}^{3}.$ Which of the following

statements are necessarily true?\medskip



(I) The span of $S$ is a plane through the origin



(II) The dimension of the span of $S$ is at most 2



(III) The span of $S$ cannot be all of $\mathbb{R}^{3}\medskip$



a) None of them$\quad$b) I and II only$\quad$c) I only$\quad$d) II and III

only$\quad$e) I, II, and III\pagebreak 



\item  Let $A$ be the matrix

\[

A=\left(

\begin{array}

[c]{rrrr}%

1 & 2 & 1 & 0\\

3 & 0 & 1 & 2\\

1 & -4 & -1 & 2

\end{array}

\right)  .

\]

Find a basis for $RS\left(  A\right)  ,$ a basis for $NS\left(  A\right)  ,$

and a basis for $CS\left(  A\right)  .$ State the dimensions of $RS\left(

A\right)  ,$ $NS\left(  A\right)  ,$ and $CS\left(  A\right)  .$\pagebreak 



\item  Suppose $U$ is a matrix in row-echelon form obtained from a matrix $A$

by means of row operations. Which of the following statements are necessarily true?\medskip



(I) $NS\left(  A\right)  =NS(U)$



(II) $RS(A)=RS(U)$



(III) $CS(A)=CS(U)$



(IV) $rk\left(  A\right)  =rk\left(  U\right)  $



(V) $\dim CS(A)=\dim CS(U)\medskip$



a) All of them$\quad\quad$b) IV and V only$\quad$c) I, II, and IV only$\quad

$d) I, II, IV, and V only$\quad$e) I and II\ only\pagebreak 



\item  Below are several examples of a vector space $V$ and a subset $S$ of

$V.$ In each case state whether or not $S$ is a subspace of $V.$ If $S$ is

\textit{not} a subspace of $V,$ explain why not.



a) $V=C\left[  0,5\right]  ,$ $S=\left\{  f\in V|f\left(  0\right)

=3\right\}  $



b) $V=M_{33},$ $S=\left\{  A|\text{ }A\text{ is invertible}\right\}  $



c) $V=\mathbb{R}^{3},$ $S=\left\{  \left(  0,t,2t\right)  |\text{ }%

t\in\mathbb{R}\right\}  $



d) $V=C\left[  a,b\right]  ,$ $S=\left\{  f\in V|f\left(  x\right)

\leq0\text{ for all }x\right\}  $\pagebreak 



\item  Determine whether the following polynomials constitute a basis for

$\mathcal{P}_{2}$:

\begin{align*}

p_{1}\left(  x\right)   & =1\\

p_{2}\left(  x\right)   & =1+x\\

p_{3}\left(  x\right)   & =1+x+x^{2}.

\end{align*}

If they do, find the coordinates of $f\left(  x\right)  =3+4x+5x^{2}$ with

respect to this basis.\pagebreak 



\item  Consider the vectors $u_{1}=\left(  1,1,0,0\right)  ,$ $u_{2}=\left(

1,0,2,1\right)  $ and $u_{3}=\left(  1,2,-2,-1\right)  $ in $\mathbb{R}^{4}.$

What is the dimension of the span of $\left\{  u_{1},u_{2},u_{3}\right\}  $?

Show your work and explain your reasoning clearly. \textit{Hint}: are the

vectors linearly independent?\pagebreak 



\item  Determine whether each of the following collections of vectors in

linearly independent. Explain your reasoning in each case.



a) $u_{1}=\left(  2,\frac{3}{2},50,\frac{5}{3}\right)  ,$ $u_{2}=\left(

23,\frac{5}{6},105,30\right)  ,$ $u_{3}=\left(  \frac{3}{7},0,22,-11\right)

,$ $u_{4}=\left(  22,0,0,9\right)  ,$ $u_{5}=\left(  1,1,1,0,-99\right)  $ in

$\mathbb{R}^{4}$



b) $w_{1}=\left(  1,2,5,6,7\right)  ,$ $w_{2}=\left(  7,6,5,2,1\right)  ,$

$w_{3}=\left(  0,0,0,0,0\right)  $ in $\mathbb{R}^{5}$\pagebreak 



\item  Consider the linear transformation $T:\mathbb{R}^{3}\rightarrow

\mathbb{R}^{2}$ given by

\[

T\left(  x,y,z\right)  =\left(  3x-z,2y+z\right)  .

\]

Find the matrix that represents $T.$ (That is, find the matrix $A$ such that

$f_{A}=T.$)\vspace{3.5in}



\item  Suppose that $A$ is a $3\times4$ matrix and that $\dim NS\left(

A\right)  =1.$ Which of the following statements are necessarily true?\medskip



(I) The rows of $A$ are linearly independent



(II) The columns of $A$ are linearly dependent



(III) The row echelon form of $A$ has a row of all zeros



(IV) The rank of $A$ is one\medskip



a) None of them$\quad$b) All of them$\quad$c) I and II only$\quad$d) I, II,

and III only$\quad$e) II only

\end{enumerate}

\end{document}