3.5.E: Problems on Linear Spaces (Exercises)
- Page ID
- 22263
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Prove that \(F^{n}\) in Example \((\mathrm{b})\) is a vector space, i.e., that it satisfies all laws stated in Theorem 1 in §§1-3; similarly for \(W\) in Example (d).
Verify that dot products in \(C^{n}\) obey the laws \((\mathrm{i})-\left(\mathrm{v}^{\prime}\right) .\) Which of these laws would fail if these products were defined by
\[
x \cdot y=\sum_{k=1}^{n} x_{k} y_{k} \text { instead of } x \cdot y=\sum_{k=1}^{n} x_{k} \overline{y}_{k} ?
\]
How would this affect the properties of absolute values given in \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right) ?\)
Complete the proof of formulas \(\left(\mathrm{a}^{\prime}\right)-\left(\mathrm{d}^{\prime}\right)\) for Euclidean spaces. What change would result if property (ii) of dot products were restated as
\[
" x \cdot x \geq 0 \text { and } \overrightarrow{0} \cdot \overrightarrow{0}=0^{\prime \prime} ?
\]
Define orthogonality, parallelism and angles in a general Euclidean space following the pattern of §§1-3 (text and Problem 7 there). Show that \(u=\overrightarrow{0}\) iff \(u\) is orthogonal to all vectors of the space.
Define the basic unit vectors \(e_{k}\) in \(C^{n}\) exactly as in \(E^{n},\) and prove
Theorem 2 in §§1-3 for \(C^{n}\left(\text { replacing } E^{1} \text { by } C\right).\) Also, do Problem 5\((\mathrm{a})\) of §§1-3 for \(C^{n}\).
Define hyperplanes in \(C^{n}\) as in Definition 3 of §§4-6, and prove Theorem 1 stated there, for \(C^{n} .\) Do also Problems \(4-6\) there for \(C^{n}\) (replacing \(E^{1}\) by \(C )\) and Problem 4 there for vector spaces in general (replacing \(E^{1}\) by the scalar field \(F ) .\)
Do Problem 3 of §§4-6 for general Euclidean spaces (real or complex). Note: Do not replace \(E^{1}\) by \(C\) in the definition of a line and a line segment.
A finite set of vectors \(B=\left\{x_{1}, \ldots, x_{m}\right\}\) in a linear space \(V\) over \(F\) is said to be independent iff
\[
\left(\forall a_{1}, a_{2}, \ldots, a_{m} \in F\right) \quad\left(\sum_{i=1}^{m} a_{i} x_{i}=\overrightarrow{0} \Longrightarrow a_{1}=a_{2}=\cdots=a_{m}=0\right).
\]
Prove that if \(B\) is independent, then
(i) \(\overrightarrow{0} \notin B\);
(ii) each subset of \(B\) is independent \((\emptyset \text { counts as independent }) ;\) and
(iii) if for some scalars \(a_{i}, b_{i} \in F\),
\[
\sum_{i=1}^{m} a_{i} x_{i}=\sum_{i=1}^{m} b_{i} x_{i},
\]
then \(a_{i}=b_{i}, i=1,2, \ldots, m\).
Let \(V\) be a vector space over \(F\) and let \(A \subseteq V .\) By the span of \(A\) in \(V\), denoted \(\operatorname{span}(A),\) is meant the set of all "linear combinations" of vectors from \(A,\) i.e., all vectors of the form
\[
\sum_{i=1}^{m} a_{i} x_{i}, \quad a_{i} \in F, x_{i} \in A, m \in N.
\]
Show that \(\operatorname{span}(A)\) is itself a vector space \(V^{\prime} \subseteq V\) (a subspace of \(V )\) over the same field \(F,\) with the operations defined in \(V .\) (We say that A spans \(V^{\prime} .\) Show that in \(E^{n}\) and \(C^{n},\) the basic unit vectors span the entire space.