Skip to main content
Mathematics LibreTexts

5.E: Exercises for Chapter 5

  • Page ID
    292
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

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

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

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

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)

    Calculational Exercises

    1. Show that the vectors \(v_1 = (1, 1, 1), v_2 = (1, 2, 3)\), and \(v_3 = (2, −1, 1)\) are linearly independent in \(\mathbb{R}^3\). Write \(v = (1, −2, 5)\) as a linear combination of \(v_1 , v_2\), and \(v_3\).

    2. Consider the complex vector space \(V = \mathbb{C}^3\) and the list \((v_1 , v_2 , v_3 )\) of vectors in \(V\) , where

    \[v_1 = (i, 0, 0),~ v_2 = (i, 1, 0),~ v_3 = (i, i, −1).\]

    (a) Prove that \(span(v_1 , v-2 , v_3 ) = V.\)
    (b) Prove or disprove: \((v_1 , v_2 , v_3)\) is a basis for \(V.\)

    3. Determine the dimension of each of the following subspaces of \(\mathbb{F}^4\) .
    (a) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = 0\}.\)
    (b) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 \}.\)
    (c) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 \}.\)
    (d) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_4 = x_1 + x_2 , x_3 = x_1 − x_2 , x_3 + x_4 = 2x_1 \}.\)
    (e) \(\{(x_1 , x_2 , x_3 , x_4 ) \in \mathbb{F}^4 | x_1 = x_2 = x_3 = x_4 \}.\)

    4. Determine the value of \(\lambda \in \mathbb{R}\) for which each list of vectors is linear dependent.
    (a) \(((\lambda, −1, −1), (−1, \lambda, −1), (−1, −1, \lambda))\) as a subset of \(\mathbb{R}^3.\)
    (b) \(sin2 (x), cos(2x), \lambda\) as a subset of \(\cal{C}(\mathbb{R}).\)

    5. Consider the real vector space \(V = \mathbb{R}^4.\) For each of the following five statements, provide either a proof or a counterexample.
    (a) \(dim V = 4.\)
    (b) \(span((1, 1, 0, 0), (0, 1, 1, 0), (0, 0, 1, 1)) = V.\)
    (c) The list \(((1, −1, 0, 0), (0, 1, −1, 0), (0, 0, 1, −1), (−1, 0, 0, 1))\) is linearly independent.
    (d) Every list of four vectors \(v_1 , \ldots, v_4 \in V\) , such that \(span(v_1 , \ldots, v_4 ) = V\) , is linearly independent.
    (e) Let \(v_1\) and \(v_2\) be two linearly independent vectors in \(V\) . Then, there exist vectors \(u, w \in V\) , such that \((v_1 , v_2 , u, w)\) is a basis for \(V.\)

    Proof-Writing Exercises

    1. Let \(V\) be a vector space over \(\mathbb{F}\) and define \(U = span(u_1, u_2, \ldots ,u_n)\), where for each
    \(i = 1, \ldots ,n, u_i \in V \). Now suppose \(v \in U\). Prove


    \[U = span(v, u_1, u_2, \ldots ,u_n) .\]

    2. Let \(V\) be a vector space over \(\mathbb{F}\), and suppose that the list \((v_1 , v_2 , . . . , v_n )\) of vectors spans \(V\) , where each \(v_i \in V\) . Prove that the list


    \[(v_1 − v_2 , v_2 − v_3 , v_3 − v_4 , \ldots , v_{n−2} − v_{n−1} , v_{n−1} − v_n , v_n )\]

    also spans \(V.\)

    3. Let \(V\) be a vector space over \(\mathbb{F}\), and suppose that \((v_1 , v_2 , \ldots, v_n)\) is a linearly independent list of vectors in \(V\) . Given any \(w \in V\) such that

    \[(v_1 + w, v_2 + w, \ldots , v_n + w)\]

    is a linearly dependent list of vectors in \(V\) , prove that \(w \in span(v_1 , v_2 , \ldots, v_n).\)

    4. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(dim(V ) = n\) for some \(n \in \mathbb{Z}_+\). Prove that there are \(n\) one-dimensional subspaces \(U_1 , U_2 , \ldots , U_n\) of \(V\) such that

    \[V = U_1 \oplus U_2 \oplus \cdots \oplus U_n .\]

    5. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\), and suppose that \(U\) is a subspace of \(V\) for which \(dim(U) = dim(V ).\) Prove that \(U = V.\)

    6. Let \(\mathbb{F}_m [z] \) denote the vector space of all polynomials with degree less than or equal to \(m \in \mathbb{Z}_+\) and having coefficient over \(\mathbb{F}\), and suppose that\( p_0 , p_1 , \ldots , p_m \in \mathbb{F}_m [z]\) satisfy \(p_j (2) = 0\). Prove that \((p_0 , p_1 , \ldots , p_m )\) is a linearly dependent list of vectors in \(\mathbb{F}_m [z].\)

    7. Let \(U\) and \(V\) be five-dimensional subspaces of \(\mathbb{R}^9\) . Prove that \(U \cap V = \{0\}.\)

    8. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F},\) and suppose that \(U_1 , U_2 , \ldots, U_m\) are any \(m\) subspaces of \(V\) . Prove that


    \[dim(U_1 + U_2 + \cdots + U_m ) \leq dim(U_1 ) + dim(U_2 ) + \cdots + dim(U_m ).\]


    This page titled 5.E: Exercises for Chapter 5 is shared under a not declared license and was authored, remixed, and/or curated by Isaiah Lankham, Bruno Nachtergaele, & Anne Schilling.

    • Was this article helpful?