4.8.E: Exercise for Section 4.6
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}2\\1\\1\\1\end{array}\right],\:\left[\begin{array}{r}-1\\0\\-1\\-1\end{array}\right],\:\left[\begin{array}{r}5\\2\\3\\3\end{array}\right],\:\left[\begin{array}{r}-1\\1\\-2\\-2\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}0\\1\\1\\-1\end{array}\right],\:\left[\begin{array}{r}-1\\-1\\-2\\2\end{array}\right],\:\left[\begin{array}{r}2\\3\\5\\-5\end{array}\right],\:\left[\begin{array}{r}0\\1\\2\\-2\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}-2\\1\\1\\-3\end{array}\right],\:\left[\begin{array}{r}-9\\4\\3\\-9\end{array}\right],\:\left[\begin{array}{r}-33\\15\\12\\-36\end{array}\right],\:\left[\begin{array}{r}-22\\10\\8\\-24\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}-1\\1\\-1\\-2\end{array}\right],\:\left[\begin{array}{r}-4\\3\\-2\\-4\end{array}\right],\:\left[\begin{array}{r}-3\\2\\-1\\-2\end{array}\right],\:\left[\begin{array}{r}-1\\1\\-2\\-4\end{array}\right],\:\left[\begin{array}{r}-7\\5\\-3\\-6\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}2\\3\\2\\1\end{array}\right],\:\left[\begin{array}{r}8\\15\\6\\3\end{array}\right],\:\left[\begin{array}{r}3\\6\\2\\1\end{array}\right],\:\left[\begin{array}{r}4\\6\\6\\3\end{array}\right],\:\left[\begin{array}{r}8\\15\\6\\3\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}0\\2\\0\\-1\end{array}\right],\:\left[\begin{array}{r}-1\\6\\0\\-2\end{array}\right],\:\left[\begin{array}{r}-2\\16\\0\\-6\end{array}\right],\:\left[\begin{array}{r}-3\\22\\0\\-8\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}5\\1\\1\\4\end{array}\right],\:\left[\begin{array}{r}14\\3\\2\\8\end{array}\right],\:\left[\begin{array}{r}38\\8\\6\\24\end{array}\right],\:\left[\begin{array}{r}47\\10\\7\\28\end{array}\right],\:\left[\begin{array}{r}10\\2\\3\\12\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(H=\mathrm{span}\left\{\left[\begin{array}{r}6\\1\\1\\5\end{array}\right],\:\left[\begin{array}{r}17\\3\\2\\10\end{array}\right],\:\left[\begin{array}{r}52\\9\\7\\35\end{array}\right],\:\left[\begin{array}{r}18\\3\\4\\20\end{array}\right]\right\}\). Find the dimension of \(H\) and determine a basis.
Let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:\sin(u_1)=1\right\}\). Is \(M\) a subspace? Explain.
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No. Let \(\vec{u}=\left[\begin{array}{c}\frac{\pi}{2} \\ 0\\0\\0\end{array}\right]\). Then \(2\vec{u}\cancel{\in}M\) although \(\vec{u}\in M\).
Let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:||u_1||\leq 4\right\}\). Is \(M\) a subspace? Explain.
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No. \(\left[\begin{array}{c}1\\0\\0\\0\end{array}\right]\in M\) but \(10\left[\begin{array}{c}1\\0\\0\\0\end{array}\right]\cancel{\in }M\).
Let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:u_1\geq 0\text{ for each }i=1,2,3,4 \right\}\). Is \(M\) a subspace? Explain.
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This is not a subspace. \(\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]\) is in it. However, \((-1)\left[\begin{array}{c}1\\1\\1\\1\end{array}\right]\) is not.
Let \(\vec{w}\), \(\vec{w}_1\) be given vectors in \(\mathbb{R}^4\) and define \[M=\left\{\vec{u}=\left[\begin{array}{c}u_1\\u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4 :\vec{w}\bullet\vec{u}=0\text{ and }\vec{w}_1\bullet\vec{u}=0\right\}.\nonumber\] Is \(M\) a subspace? Explain.
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This is a subspace because it is closed with respect to vector addition and scalar multiplication.
Let \(\vec{w}\in\mathbb{R}^4\) and let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:\vec{w}\bullet\vec{u}=0\right\}\). Is \(M\) a subspace? Explain.
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Yes, this is a subspace because it is closed with respect to vector addition and scalar multiplication.
Let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:u_3\geq u_1\right\}\). Is \(M\) a subspace? Explain.
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This is not a subspace. \(\left[\begin{array}{c}0\\0\\1\\0\end{array}\right]\) is in it. However \((-1)\left[\begin{array}{c}0\\0\\1\\0\end{array}\right]=\left[\begin{array}{r}0\\0\\-1\\0\end{array}\right]\) is not.
Let \(M=\left\{\vec{u}=\left[\begin{array}{c}u_1 \\ u_2\\u_3\\u_4\end{array}\right]\in\mathbb{R}^4:u_3=u_1=0\right\}\). Is \(M\) a subspace? Explain.
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This is a subspace. It is closed with respect to vector addition and scalar multiplication.
Consider the set of vectors \(S\) given by \[S=\left\{\left[\begin{array}{c}4u+v-5w \\ 12u+6v-6w \\ 4u+4v+4w\end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is \(S\) a subspace of \(\mathbb{R}^3\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the set of vectors \(S\) given by \[S=\left\{\left[\begin{array}{c}2u+6v+7w \\ -3u-9v-12w \\ 2u+6v+6w \\ u+3v+3w \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is \(S\) a subspace of \(\mathbb{R}^4\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the set of vectors \(S\) given by \[S=\left\{\left[\begin{array}{c}2u+v \\ 6v-3u+3w \\ 3v-6u+3w \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is this set of vectors a subspace of \(\mathbb{R}^3\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the vectors of the form \[\left\{\left[\begin{array}{c}2u+v+7w \\ u-2v+w \\ -6v-6w \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is this set of vectors a subspace of \(\mathbb{R}^3\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the vectors of the form \[\left\{\left[\begin{array}{c}3u+v+11w \\ 18u+6v+66w \\ 28u+8v+100w \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is this set of vectors a subspace of \(\mathbb{R}^3\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the vectors of the form \[\left\{\left[\begin{array}{c}3u+v \\ 2w-4u \\ 2w-2v-8u \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is this set of vectors a subspace of \(\mathbb{R}^3\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the set of vectors \(S\) given by \[\left\{\left[\begin{array}{c}u+v+w \\ 2u+2v+4w \\ u+v+w \\ 0 \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is \(S\) is a subspace of \(\mathbb{R}^4\)? If so, explain why, give a basis for the subspace and find its dimension.
Consider the set of vectors \(S\) given by \[\left\{\left[\begin{array}{c}v \\ -3u-3w \\ 8u-4v+4w \end{array}\right] :u,v,w\in\mathbb{R}\right\}.\nonumber\] Is \(S\) is a subspace of \(\mathbb{R}^4\)? If so, explain why, give a basis for the subspace and find its dimension.
If you have \(5\) vectors in \(\mathbb{R}^5\) and the vectors are linearly independent, can it always be concluded they span \(\mathbb{R}^5\)? Explain.
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Yes. If not, there would exist a vector not in the span. But then you could add in this vector and obtain a linearly independent set of vectors with more vectors than a basis.
If you have \(6\) vectors in \(\mathbb{R}^5\), is it possible they are linearly independent? Explain.
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They can't be.
Suppose \(A\) is an \(m\times n\) matrix and \(\{\vec{w}_1,\cdots ,\vec{w}_k\}\) is a linearly independent set of vectors in \(A(\mathbb{R}^n ) ⊆ \mathbb{R}^m\). Now suppose \(A\vec{z}_i = \vec{w}_i\). Show \(\{\vec{z}_1 ,\cdots ,\vec{z}_k\}\) is also independent.
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Say \(\sum\limits_{i=1}^k c_i\vec{z}_i=\vec{0}\). Then apply \(A\) to it as follows. \[\sum\limits_{i=1}^k c_aA\vec{z}_i=\sum\limits_{i=1}^kc_i\vec{w}_i=\vec{0}\nonumber\] and so, by linear independence of the \(\vec{w}_i\), it follows that each \(c_i=0\).
Let \(V\) and \(W\) be subspaces of \(\mathbb{R}^n\). Show that the intersection \(V \cap W\) is also a subspace of \(\mathbb{R}^n\)