4.E: Exercises for Chapter 4
- Page ID
- 318
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Calculational Exercises
1. For each of the following sets, either show that the set is a vector space or explain why it is not a vector space.
(a) The set \(\mathbb{R}\) of real numbers under the usual operations of addition and multiplication.
(b) The set \(\{(x, 0)~ |~ x \in \mathbb{R}\}\) under the usual operations of addition and multiplication on \(\mathbb{R}^2.\)
(c) The set \(\{(x, 1) ~|~ x \in \mathbb{R}\}\) under the usual operations of addition and multiplication on \(\mathbb{R}^2.\)
(d) The set \(\{(x, 0) ~| ~x \in \mathbb{R}, x \geq 0\}\) under the usual operations of addition and multiplication on \(\mathbb{R}^2.\)
(e) The set \(\{(x, 1)~ |~ x \in \mathbb{R}, x \geq 0\}\) under the usual operations of addition and multiplication on \(\mathbb{R}^2.\)
(f) The set \(\left\{ \left[ \begin{array}{cc} a & a+b \\ a+b & a \end{array} \right] ~|~ a, b \in \mathbb{R} \right\} \) under the usual operations of addition and multiplication on \(\mathbb{R}^{2 \times 2}.\)
(g) The set \(\left\{ \left[ \begin{array}{cc} a & a+b+1 \\ a+b & a \end{array} \right] ~|~ a, b \in \mathbb{R} \right\} \) under the usual operations of addition and multiplication on \(\mathbb{R}^{2 \times 2}.\)
under the usual operations of addition
2. Show that the space \(V = \{(x_1 , x_2 , x_3 ) \in \mathbb{F}^3 ~|~ x_1 + 2x_2 + 2x_3 = 0\}\) forms a vector space.
3. For each of the following sets, either show that the set is a subspace of \(\cal{C}(\mathbb{R})\) or explain why it is not a subspace.
(a) The set \(\{f \in \cal{C}(\mathbb{R}) ~|~ f (x) \leq 0, \forall x \in \mathbb{R}\}.\)
(b) The set \(\{f \in \cal{C}(\mathbb{R}) ~|~ f(0) = 0\}. \)
(c) The set \(\{f \in \cal{C}(\mathbb{R}) ~|~ f (0) = 2\}.\)
(d) The set of all constant functions.
(e) The set \(\{\alpha + \beta sin(x) ~|~ \alpha, \beta \in \mathbb{R}\}.\)
4. Give an example of a nonempty subset \(U \subset \mathbb{R}^2\) such that \(U\) is closed under scalar multiplication but is not a subspace of \(\mathbb{R}^2.\)
5. Let \(\mathbb{F}[z]\) denote the vector space of all polynomials having coefficient over \(\mathbb{F}\), and define \(U\) to be the subspace of \(\mathbb{F}[z]\) given by
\[U = \{az^2 + bz^5 ~|~ a, b \in \mathbb{F}\}.\]
Find a subspace \(W\) of \(\mathbb{F}[z]\) such that \(\mathbb{F}[z] = U \oplus W .\)
Proof-Writing Exercises
1. Let \(V\) be a vector space over \(\mathbb{F}\). Then, given \(a \in \mathbb{F}\) and \(v \in V\) such that \(av = 0\), prove that either \(a = 0\) or \(v = 0.\)
2. Let \(V\) be a vector space over \(\mathbb{F},\) and suppose that \(W_1\) and \(W_2\) are subspaces of \(V.\)
Prove that their intersection \(W_1 \cap W_2\) is also a subspace of \(V.\)
3. Prove or give a counterexample to the following claim:
Claim. Let \(V\) be a vector space over \(\mathbb{F},\) and suppose that \(W_1, W_2,\) and \(W_3\) are subspaces of \(V\) such that \(W_1 + W_3 = W_2 + W_3.\) Then \(W_1 = W_2.\)
4. Prove or give a counterexample to the following claim:
Claim. Let \(V\) be a vector space over \(\mathbb{F},\) and suppose that \(W_1 , W_2,\) and \(W_3\) are subspaces of \(V\) such that \(W_1 \oplus W_3 = W_2 \oplus W_3.\) Then \(W_1 = W_2.\)
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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