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{\bf \centerline
{Math 261 -- Linear Algebra}} 
{\bf \centerline{Fall 2000 (Exam 1)}}


\bigskip

\noindent{\bf 1.} {\sl (15 points)} Let $W_1=\{(a,b)\in \Bbb{R}^2\mid
a+2b=0\}$ and $W_2=\{(a,b)\in \Bbb{R}^2\mid
a+2b=1\}$.  Decide which  of $W_1$ and $W_2$ are subspaces of $\Bbb{R}^2$.
Prove your answer.\bigskip
 
\noindent{\bf 2.} {\sl (15 points)} Consider the two subsets $A$ and $B$ of $\Bbb{R}^3$ given
by
$$A=\{(1,1,0)\;, (0,1,1)\},\; \;B=\{(1,2,1), (1,0,-1), (3,4,1)\}.$$
\noindent  Show that the subspaces of $\Bbb {R}^3$ spanned by $A$ and $B$ are the same.
 

\bigskip 

\noindent{\bf 3.} {\sl (15 points)} Consider the subspace $W$ of $\Bbb R^4$ spanned by 
the vectors $$(1,0,0, 3),\;
(-1,-2,0,1),\; (1,2,3,-2).$$
Is $(2,3,-1,5)\in W$?

\vskip10pt
\noindent{\bf 4.} {\sl (15 points)} Is the set $\{ (1,-1,1), (1,1,1),(1,0,1)\}\subseteq \Bbb 
R^3$ linearly
independent? If not, find a basis for the subspace of $\Bbb{R}^3$ spanned by this set.
\bigskip
\noindent{\bf 5.} {\sl (15 points)} Let $F$ be a field. Consider the subset
$$W=\{(\lambda_1,\ldots, \lambda_n)\in F^n\mid  \lambda_1+\ldots +\lambda_n=0\}$$ of $F^n$,
which you may assume  
  is  a subspace of $F^n$. Find a basis for $W$ and determine the dimension of $W$ as a
$F$-vector space.\bigskip   
\noindent{\bf 6.} {\sl (25 points)} 
 \noindent{\bf (a)} Carefully state the definition of the span of a subset $S$ of a $F$-vector
space $V$, and the definition of  linear  independence of $S$.\medskip

\noindent{\bf (b)}
Let $V$ be a vector space and
$\{v_1,\ldots,v_n\}$ be a set of
$n$ 
 vectors in $V$. Fix an integer $i$ with $1\leq i\leq n-1$. Prove that $\{v_1,\ldots,v_n\}$ is
linearly independent if and only if the following three conditions all hold:
$A=\{v_1,\ldots,v_i\}$ is linearly independent, 
$B=\{v_{i+1},\ldots, v_n\}$ is linearly independent and $\text{\rm span}(A)\cap \text{\rm
span}(B)=\{0\}$. 


%\noindent{\bf 4.} Let $V$ be an n-dimensional vector space and $A\subseteq V$ a
%set with $n$ elements. Suppose that $\Cal L \{A\}=V$. Must $A$ be a 
%basis for $V$?
%Either prove that the answer is \it yes \rm or give a counterexample.
%\vskip10pt
%
%\noindent{\bf 5.} Let $V$ be a vector space and consider the \it diagonal \rm set
%$\Delta \subseteq V\times V\times ...V$ ($p$ times) given by
%$$\Delta =\{(v,v,...,v) \;|v\in V\}.$$
%\noindent{\bf (i)} Show that $\Delta$ is a subspace of $V\times V\times ...V$.
%
%\noindent{\bf(ii)} If $\dim V=n$, what is $\dim \Delta$?
%\vskip10pt
%
%\noindent{\bf 6.} Define what it means for a vector space $V$ to be the 
%direct sum of
%non-trivial subspaces $W_1$ and $W_2$. For each $k, \;0<k<n^2$, 
%exhibit the space $M^{n\times n}$
%of $n\times n$ matrices as a direct sum where one of the subspaces 
%has dimension $k$.
%\vskip10pt
%
%\noindent{\bf 7.} Let $W_1$ and $W_2$ be subspaces of the vector space 
%$V$ such that the union  $W_1\cup W_2$ is
%also a subspace. Show that one of the subspaces is contained in the other 
%(i.e. either $W_1\subseteq W_2$ or $W_2\subseteq W_1$).
%\vskip10pt

%\noindent{\bf 8.} Give infinitely many distinct examples of infinite - 
%dimensional vector spaces.
\vskip10pt
\vfill\eject\bye
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