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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 1 -- More Examples of Subspaces}
\centerline{January 28, 2004}
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{\it Background}
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Recall that we have said a subset $W$ of a vector space $V$
is a ({\it vector}) {\it subspace} if $W$ is itself a vector
space under the vector sum and scalar multiplication operations
from $V$.  We also saw the following theorem that gives a
``shortcut test'' for deciding whether a given $W \subseteq V$
is a subspace:

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{\bf Theorem.} Let $V$ be a vector space.
 A nonempty subset $W \subset V$ is a subspace
if and only if whenever ${\bf x},{\bf y} \in W$ and $c\in {\bf R}$,
then $c{\bf x} + {\bf y} \in W$ (using the vector sum and scalar
multiplication operations from $V$).

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\noindent
Today, in our first discussion, we will get some additional 
practice using the Theorem to decide whether given subsets are
subspaces.
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{\it Discussion Questions}
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\item{A)}  For each of the given vector spaces $V$ and subsets
$W \subset V$, using the test given by the 
theorem above, decide whether or not $W$ is a subspace of $V$.
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\itemitem{1)}  $V = {\bf R}^2$ $W = \{(x,y) : y = 12 x^2\}$
\itemitem{2)}  $V = {\bf R}^3$, $W = \{(x,y,z) : 2x + 3y - z = 0\ {\rm and}\ 
x - y + z = 0\}$.  Also, what is the set $W$ in geometric terms?
\itemitem{3)}  $V = {\bf R}^4$, $W = \{(x,y,z,w) : x - y + 2z + w = 1\}$
\itemitem{4)}  $V = F({\bf R})$ 
(that is, the vector space of all functions $f : {\bf R} \to {\bf R}$
and 
$$W = \{ f\in F({\bf R}) : f(x + 1) = f(x) {\rm\ for\ all\ } x\}.$$
Also, give an example of a nonconstant function in $W$.
\itemitem{5)}  $V = P_5({\bf R})$ (the vector space of polynomials
of degree $\le 5$ with real coefficients), 
$W = \{ p \in V : x^2 p''(x) =  3x p'(x) - 3 p(x)\}$.  
Also find an example of a nonzero polynomial in $W$.  Hint:
A good place to start is to determine for which exponents $n$
we have $x^n \in W$.
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\item{B)}  We saw in class that the {\it intersection} of any 
collection of subspaces of a vector space is a subspace.
Is the same true for {\it unions} of subspaces?
(Recall $W_1 \cup W_2 = \{{\bf x} : {\bf x} \in W_1 {\rm\ or\ } {\bf x} \in W_2\}$.)
{\it Suggestion:}  Look at some examples in $V = {\bf R}^3$ or ${\bf R}^2$.  
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{\it Assignment}
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Group writeups due in class, Monday, February 2.
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