13.E: Exercises
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Determine whether the given sets of vectors are linearly dependent or linearly independent. If they are linearly dependent, write one as a linear combination of the others.
a.) \(\vec u = (0,2), \vec v = (0,3)\)
b.) \(\vec u = (1,2,4), \vec v = (1,-2,3), \vec w = (-2,0,1)\)
c.) \(\vec u = (7,5), \vec v = (1,2), \vec w = (3,-1)\)
d) \(\vec u = (1,2,3), \vec v = (2,4,6), \vec w = (4,1,-4)\)
- Answer
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a.) linearly dependent
b.) linearly independent
c.) linearly dependent
d.) linearly dependent
Exercise \(\PageIndex{2}\)
Are the following sets subspaces of \(\mathbb{R}^3\) or not? If not, give an explanation and if so, prove it.
a.) \(V\) is the set of all \( (x,y,z)\) such that \(x=0\)
b.) \(V\) is the set of all \( (x,y,z)\) such that \(2x=3y\)
c.) \(V\) is the set of all \( (x,y,z)\) such that \(x=6\)
d.) \(V\) is the set of all \( (x,y,z)\) such that \(x+y=0\)
e.) \(V\) is the set of all \( (x,y,z)\) such that \(x+y=2\)
- Answer
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a.) yes
b.) yes
c.) no
d.) yes
e.) no
Exercise \(\PageIndex{3}\)
Is \(\vec v = (1,0,-1)\) in the span of \(\{ (5,3,4), (3,2,5)\}\)? If so, write \(\vec v\) as a linear combination of the two vectors.
Exercise \(\PageIndex{4}\)
Prove that if a finite set of vectors contains the zero vector, then that set is linearly dependent.
- Answer
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Hint: To prove a set of vectors is linearly dependent, you need to show that one of the vectors is a linear combination of the others.
Exercise \(\PageIndex{5}\)
Do the following sets of vectors form a basis for \(\mathbb{R}^n\)? Why or why not?
a.) \(\vec v_1 = (4,7), \vec v_2 = (5,6)\)
b.) \(\vec v_1 = (1,-3), \vec v_2 = (-3,9)\)
c.) \(\vec v_1 = (4,7,4), \vec v_2 = (5,6,0), \vec v_3 = (2,-1,1)\)
d.) \(\vec v_1 = (4,-1,4), \vec v_2 = (5,2,0)\)
e.) \(\vec v_1 = (4,7,4), \vec v_2 = (5,6,0), \vec v_3 = (2,-1,1), \vec v_4 = (0,1,2)\)
- Answer
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