Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

9.3: Review Problems

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    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.