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\title
Math 261 -- Linear Algebra\vskip 10pt
Fall 1999 (Exam 1)
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\noindent (1) Consider the subsets of $\Bbb R^3$ given by $$A=\{(1,1,0)\;,
(0,1,1)\},\; \;B=\{(1,2,1), (1,0,-1), (3,4,1)\}.$$
Show that the subspaces generated by $A$ and $B$ are the same.
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\noindent (2) Consider the subspace $W$ of $\Bbb R^4$ generated by 
the vectors $$(1,0,0, 3),\;
(-1,-2,0,1),\; (1,2,3,-2).$$
Does $(2,3,-1,5)\in W$?
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\noindent (3) Is the set $\{ (1,0,1), (0,1,1),(1,1,2)\}\subset \Bbb 
R^3$ linearly
independent?
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\noindent (4) Let $V$ be an n-dimensional vector space and $A\subset V$ 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.
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\noindent (5) Let $V$ be a vector space and consider the \it diagonal \rm set
$\Delta \subset V\times V\times ...V$ ($p$ times) given by
$$\Delta =\{(v,v,...,v) \;|v\in V\}.$$
(i) Show that $\Delta$ is a subspace of $V\times V\times ...V$.

\noindent(ii) If $\dim V=n$, what is $\dim \Delta$?
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\noindent (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 $k$.
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\noindent (7) Let $W_1$ and $W_2$ be subspaces of the vector space 
$V$ such that $W_1\cup W_2$ is
also a subspace. Show that one of the subspaces is contained in the other.
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\noindent (8) Give infinitely many distinct examples of infinite - 
dimensional vector spaces.
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