1.1.E: Introduction to Rⁿ (Exercises)
- Page ID
- 31488
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Let \(\mathbf{x}=(1,2), \mathbf{y}=(2,3),\) and \(\mathbf{z}=(-2,4) .\) For each of the following, plot the points \(\mathbf{x}, \mathbf{y}, \mathbf{z},\) and the indicated point \(\mathbf{w}\).
(a) \(\mathbf{w}=\mathbf{x}+\mathbf{y}\)
(b) \(\mathbf{w}=2 \mathbf{x}-\mathbf{y}\)
(c) \(\mathbf{w}=\mathbf{z}-2 \mathbf{x}\)
(d) \(\mathbf{w}=3 \mathbf{x}+2 \mathbf{y}-\mathbf{z}\)
Exercise \(\PageIndex{2}\)
Let \(\mathbf{x}=(1,3,-1), \mathbf{y}=(3,2,1),\) and \(\mathbf{z}=(-2,4,-2) .\) Compute each of the following.
(a) \(\mathbf{x}+\mathbf{y}\)
(b) \(\mathbf{x}-\mathbf{z}+3 \mathbf{y}\)
(c) \(3 \mathbf{z}-2 \mathbf{y}\)
(d) \(-3 \mathbf{x}+4 \mathbf{z}\)
- Answer
-
(a) (4,5,0)
(b) (12,5,4)
(c) (-12,8,-8)
(d) (-11,7,-5)
Exercise \(\PageIndex{3}\)
Let \(\mathbf{x}=(1,-1,2,3), \mathbf{y}=(-2,3,1,-2),\) and \(\mathbf{z}=(2,1,3,-4) .\) Compute each of the following.
(a) \(\mathbf{x}-2 \mathbf{z}\)
(b) \(\mathbf{y}+\mathbf{x}-3 \mathbf{z}\)
(c) \(-3 \mathbf{y}-\mathbf{x}+4 \mathbf{z}\)
(d) \(\mathbf{x}+3 \mathbf{z}-4 \mathbf{y}\)
Exercise \(\PageIndex{4}\)
Let \(\mathbf{x}=(1,2)\) and \(\mathbf{y}=(-2,3) .\) Compute each of the following.
(a) \(\|\mathbf{x}\|\)
(b) \(\|\mathbf{x}-\mathbf{y}\|\)
(c) \(\|3 \mathbf{x}\|\)
(d) \(\|-4 \mathbf{y}\|\)
Exercise \(\PageIndex{5}\)
Let \(x=(2,3,-1), y=(2,-1,5),\) and \(z=(3,-1,-2) .\) Compute each of the following.
(a) \(\|\mathbf{x}\|\)
(b) \(\|\mathbf{x}+2 \mathbf{y}\|\)
(c) \(\|-5 \mathbf{x}\|\)
(d) \(\|\mathbf{x}+\mathbf{y}+\mathbf{z}\|\)
- Answer
-
(a) \(\sqrt{14}\)
(b) \(\sqrt{118}\)
(c) \(5 \sqrt{14}\)
(d) \(3 \sqrt{6}\)
Exercise \(\PageIndex{6}\)
Find the distances between the following pairs of points.
(a) \(\mathbf{x}=(3,2), \mathbf{y}=(-1,3)\)
(b) \(\mathbf{x}=(1,2,1), \mathbf{y}=(-2,-1,3)\)
(c) \(\mathbf{x}=(4,2,1,-1), \mathbf{y}=(1,3,2,-2)\)
(d) \(\mathbf{z}=(3,-3,0), \mathbf{y}=(-1,2,-5)\)
(e) \(\mathbf{w}=(1,2,4,-2,3,-1), \mathbf{u}=(3,2,1,-3,2,1)\)
- Answer
-
(a) \(\sqrt{17}\)
(b) \(\sqrt{22}\)
(c) \(2 \sqrt{3}\)
(d) \(\sqrt{66}\)
(e) \(\sqrt{19}\)
Exercise \(\PageIndex{7}\)
Draw a picture of the following sets of points in \(\mathrm{R}^{2}\).
(a) \(S^{1}((1,2), 1)\)
(b) \(B^{2}((1,2), 1)\)
(c) \(\overline{B}^{2}((1,2), 1)\)
Exercise \(\PageIndex{8}\)
Draw a picture of the following sets of points in \(\mathbb{R}\).
(a) \(S^{0}(1,3)\)
(b) \(B^{1}(1,3)\)
(c) \(\overline{B}^{1}(1,3)\)
Exercise \(\PageIndex{9}\)
Describe the differences between \(S^{2}((1,2,1), 1), B^{3}((1,2,1), 1),\) and \(\overline{B}^{3}((1,2,1), 1)\) in \(\mathbb{R}^{3}\).
Exercise \(\PageIndex{10}\)
Is the point \((1,4,5)\) in the the open ball \(B^{3}((-1,2,3), 4) ?\)
Exercise \(\PageIndex{11}\)
Is the point \((3,2,-1,4,1)\) in the open ball \(B^{5}((1,2,-4,2,3), 3) ?\)
- Answer
-
No
Exercise \(\PageIndex{12}\)
Find the length and direction of the following vectors.
(a) \(\mathbf{x}=(2,1)\)
(b) \(\mathbf{z}=(1,1,-1)\)
(c) \(\mathbf{x}=(-1,2,3)\)
(d) \(\mathbf{w}=(1,-1,2,-3)\)
- Answer
-
(a) \(\|\mathbf{x}\|=\sqrt{5}\), Direction: \(\|\mathbf{u}\|=\frac{1}{\sqrt{5}}(2,1)\)
(b) \(\|\mathbf{z}\|=\sqrt{3}\), Direction: \(\|\mathbf{u}\|=\frac{1}{\sqrt{3}}(1,1,-1)\)
(c) \(\|\mathbf{x}\|=\sqrt{14}\), Direction: \(\|\mathbf{u}\|=\frac{1}{\sqrt{14}}(-1,2,3)\)
(d) \(\|\mathbf{w}\|=\sqrt{15}\), Direction: \(\|\mathbf{u}\|=\frac{1}{\sqrt{15}}(1,-1,2,-3)\)
Exercise \(\PageIndex{13}\)
Let \(\mathbf{x}=(1,3), \mathbf{y}=(4,1),\) and \(\mathbf{z}=(2,-1) .\) Plot \(\mathbf{x}, \mathbf{y},\) and \(\mathbf{z} .\) Also, show how to obtain each of the following geometrically.
(a) \(\mathbf{w}=\mathbf{x}+\mathbf{y}\)
(b) \(\mathbf{w}=\mathbf{y}-\mathbf{x}\)
(c) \(\mathbf{w}=3 \mathbf{z}\)
(d) \(\mathbf{w}=-2 \mathbf{z}\)
(e) \(\mathbf{w}=\frac{1}{2} \mathrm{z}\)
(f) \(\mathbf{w}=\mathbf{x}+\mathbf{y}+\mathbf{z}\)
(g) \(\mathbf{w}=\mathbf{x}+3 \mathbf{z}\)
(h) \(\mathbf{w}=\mathbf{x}-\frac{1}{4} \mathbf{y}\)
Exercise \(\PageIndex{14}\)
Suppose \(\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right), \mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right),\) and \(\mathbf{z}=\left(z_{1}, z_{2}, \ldots, z_{n}\right)\) are vectors in \(\mathbb{R}^{n}\) and \(a, b,\) and \(c\) are scalars. Verify the following.
(a) \(\mathbf{x}+\mathbf{y}=\mathbf{y}+\mathbf{x}\)
(b) \(\mathbf{x}+(\mathbf{y}+\mathbf{z})=(\mathbf{x}+\mathbf{y})+\mathbf{z}\)
(c) \(a(\mathbf{x}+\mathrm{y})=a \mathrm{x}+a \mathrm{y}\)
(d) \((a+b) \mathbf{x}=a \mathbf{x}+b \mathbf{x}\)
(e) \(a(b \mathbf{x})=(a b) \mathbf{x}\)
(f) \(\mathbf{x}+\mathbf{0}=\mathbf{x}\)
(g) \(1 \mathbf{x}=\mathbf{x}\)
(h) \(\mathbf{x}+(-\mathbf{x})=0,\) where \(-\mathbf{x}=-1 \mathbf{x}\)
Exercise \(\PageIndex{15}\)
Let \(\mathbf{u}=(1,1)\) and \(\mathbf{v}=(-1,1)\) be vectors in \(\mathbb{R}^{2}\)
(a) Let \(\mathbf{x}=(2,1) .\) Find scalars \(a\) and \(b\) such that \(\mathbf{x}=a \mathbf{u}+b \mathbf{v} .\) Are \(a\) and \(b\) unique?
(b) Let \(\mathbf{x}=(x, y)\) be an arbitrary vector in \(\mathbb{R}^{2} .\) Show that there exist unique scalars \(a\) and \(b\) such that \(\mathbf{x}=a \mathbf{u}+b \mathbf{v}\).
(c) The result in (b) shows that u and v form a basis for \(\mathbb{R}^{2}\) which is different from the standard basis of \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2} .\) Show that the vectors \(\mathbf{u}=(1,1)\) and \(\mathbf{w}=(-1,-1)\) do not form a basis for \(\mathbb{R}^{2} .\) (Hint: Show that there do not exist scalars \(a\) and \(b\) such that \(\mathbf{x}=a \mathbf{u}+\mathbf{w} \text { when } \mathbf{x}=(2,1).)\)
- Answer
-
(a) \(a=\frac{3}{2}, b=-\frac{1}{2}\); Yes, \(a\) and \(b\) are unique.
(b) \(a=\frac{x+y}{2}, b=\frac{y-x}{2}\)