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Mathematics LibreTexts

9.3: Review Problems

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.