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# 9.3: Review Problems

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1.    (Subspace Theorem) Suppose that $$V$$ is a vector space and that $$U \subset V$$ is a subset of $$V$$.  Check all the vector space requirements to show that
$\mu u_{1} + \nu u_{2} \in U \textit{ for all } u_{1}, u_{2} \in U, \mu, \nu \in \Re$
implies that $$U$$ is a subspace of $$V$$.

2.    (Subspaces spanning sets polynolmial span) Determine if  $$P_{3}^{\mathbb{R}}$$ be the vector space of polynomials of degree 3 or less in the variable $$x$$.
$x-x^{3} \in span\{ x^{2}, 2x+x^{2}, x+x^{3} \}.$

3.    (UandV) Let $$U$$ and $$W$$ be subspaces of $$V$$.  Are:
a)    $$U\cup W$$
b)    $$U\cap W$$
also subspaces?  Explain why or why not.  Draw examples in $$\Re^{3}$$.

4.    Let $$L:\mathbb{R}^{3}\to \mathbb{R}^3$$ where $$L(x,y,z)=(x+2y+z,2x+y+z,0)\, .$$
Find $${\rm ker} L$$, $${\rm im} L$$ and eigenspaces $$\mathbb{R}_{-1}$$, $$\mathbb{R}_{3}$$. Your answers should be subsets of $$\mathbb {R}^{3}$$.  Express them using the $${\rm span}$$ notation.