\documentclass[11pt]{article} % 

\usepackage{amsfonts,amstext}
\usepackage{amsthm}

\setlength{\oddsidemargin}{-0.15in}
\setlength{\topmargin}{-0.5in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9in}

\newcommand{\cP}{{\cal P}}
\newcommand{\IN}{{\bf N}}
\newcommand{\IZ}{{\bf Z}}
\newcommand{\IR}{{\bf R}}
\newcommand{\IQ}{{\bf Q}}
\newcommand{\points}[1]{{\it (#1 Points)}}


\begin{document} 
%\maketitle

\noindent
{\Large Math 211 \quad Homework 8
\hfill Sample Texfile}

\medskip
\noindent{\bf 1.}
Find an explicit description of Nul$A$ where 
$$A=\left[\matrix{1&-3&2&0\cr 0&0&3&0}\right].$$

[Find vectors such that Nul$A$ is the set of linear combinations of the vectors.]


\medskip
\noindent{\bf 2.}
Let
$$A=\left[\matrix{10&-8&-2&-2\cr 0&2&2&-2\cr 1&-1&6&0\cr 1&1&0&-2}\right],
\quad {\bf w}=\left[\matrix{2\cr 2\cr 0\cr 2}\right].$$
\begin{enumerate}
  \item Determine whether ${\bf w}$ is in Col$A$. 

 
  \item Determine whether ${\bf w}$ is in Nul$A$. 

\end{enumerate}

\medskip
\noindent{\bf 3.}
Let  $T:\mathbb{P}_2(t)\rightarrow \mathbb{R}^2$ be a linear transformation
defined as $T({\bf p})=\left[\matrix{{\bf p}(0)\cr {\bf p}(0)}\right]$. 
Find two polynomials 

in $\mathbb{P}_2(t)$ that span the
kernel of $T$, and describe the range of $T$. 

\medskip
[Hint: Determine the matrix $A$ of transformation of $T$. Find the kernel and 
range space using 

Nul$A$ and Col$A$.]

\medskip
\noindent{\bf 4.} 
Let $A = \left[ \matrix{-2 & 4 & -2 & -4\cr 
 2 & -6 & -3 & 1 \cr
 -3& 8 & 2 & -3\cr} \right]$. 
Fine bases for Nul$A$ and Col$A$. 




\medskip
\noindent{\bf 5.}
Find a basis for the space spanned by the following vectors.
$$\left[\matrix{1 \cr 0 \cr 0 \cr 1\cr}\right], \ 
\left[\matrix{-2 \cr 0 \cr 0 \cr 2\cr}\right], \ 
\left[\matrix{3 \cr -1 \cr 1 \cr -1\cr}\right], \ 
\left[\matrix{5 \cr -3 \cr 3 \cr -4\cr}\right], \ 
\left[\matrix{2 \cr -1 \cr 1 \cr 0\cr}\right].$$



\medskip
\noindent{\bf 6.}
Let 
$${\bf v_1} = \left[\matrix{3 \cr 4 \cr -2 \cr -5\cr}\right], \ 
{\bf v_2} = \left[\matrix{4 \cr 3 \cr 2 \cr 4\cr}\right], \ 
{\bf v_3} = \left[\matrix{2 \cr 5  \cr -6 \cr -14\cr}\right].$$

It can be verified that $2{\bf v_1}-{\bf v_2}-{\bf v_3}
={\bf 0}$. Find a basis for $H = {\rm Span}\{{\bf v_1}, {\bf v_2},
{\bf v_3}\}.$

\medskip\noindent{\bf 7.} Find a basis for the vector space $H$ of 
continuous functions spanned by the set 
$$\{\cos t, \sin t, \sin 2t, \sin t\cos t\}.$$
 

\end{document}
