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Find a basis of the image of the matrix $\mathbf{\left[}\begin{array}{cccccc}\mathbf{0}& \mathbf{1}& \mathbf{2}& \mathbf{0}& \mathbf{0}& \mathbf{3}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{4}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{5}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}$.

In Exercises 1through 20, find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image ofrole="math" localid="1664275567164" $\mathit{A}$and a basis of the kernel of$\mathit{A}$.

$\left(\begin{array}{cc}1& -3\\ 2& -6\\ 3& -9\end{array}\right)$

In Exercises 1through 20, find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image of$\mathit{A}$and a basis of the kernel of $\mathit{A}$.

$\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right)$

In Exercises1through 20, find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image of$\mathit{A}$and a basis of the kernel of role="math" localid="1664259082494" $\mathit{A}$

role="math" localid="1664259029752" $\left(\begin{array}{cc}1& 3\\ 2& 6\end{array}\right)$

In Exercises 1through 20, find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image of$\mathbf{A}$and a basis of the kernel of $\mathit{A}$.

$\left(\begin{array}{cc}1& 4\\ 2& 8\end{array}\right)$

In Exercises$\mathbf{1}$through$\mathbf{20}$, find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image of$\mathit{A}$and a basis of the kernel of $\mathit{A}$.

$\left(\begin{array}{ccc}1& -2& 3\\ 2& 4& 6\end{array}\right)$

Find all points$\mathit{P}$ in the plane such that you can fit infinitely many cubics through the points

$\left(0,0\right)\mathbf{,}\left(1,0\right)\mathbf{,}\left(2,0\right)\mathbf{,}\left(3,0\right)\mathbf{,}\left(0,1\right)\mathbf{,}\left(0,2\right)\mathbf{,}\left(0,3\right)\mathbf{,}\left(1,1\right)\mathbf{,}\mathit{P}$.

Use Exercise 67 to construct a basis of ${\mathbf{鈩�}}^{\mathbf{4}}$that consists of the vectors

$\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{2}\\ \mathbf{3}\\ \mathbf{4}\end{array}\mathbf{\right]}\mathbf{,}\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{4}\\ \mathbf{6}\\ \mathbf{8}\end{array}\mathbf{\right]}$

And some of the vectors${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{2}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{3}}$and${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_4$in ${\mathbf{鈩�}}^{\mathbf{4}}$.

Consider two subspaces $\mathit{V}$and $\mathit{W}$of ${\mathbf{鈩�}}^{\mathbf{n}}$. Show that $\mathbf{dim}\left(V\right)\mathbf{+}\mathbf{dim}\left(W\right)\mathbf{=}\mathbf{dim}\left(V鈭�W\right)\mathbf{+}\mathbf{dim}\left(V+W\right)$. For the definition of $\mathbf{V}\mathbf{+}\mathbf{W}$, see Exercise 3.2.50. Hint: Pick a basis ${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}$of $\mathbf{V}\mathbf{鈭�}\mathbf{W}$. Using Exercise 67, construct bases ${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{p}}$of $\mathbf{V}$and ${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{q}}$of $\mathit{W}$. Show that${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}}_{\mathbf{p}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{w}}}_{\mathbf{q}}$ is a basis of $\mathit{V}\mathbf{+}\mathit{W}$. Demonstrating linear independence is somewhat challenging.

In Exercises 1through 20, , find the redundant column vectors of the given matrix $\mathit{A}$鈥渂y inspection.鈥� Then find a basis of the image of $\mathit{A}$ and a basis of the kernel of $\mathit{A}$.

$\mathbf{\left(}\begin{array}{ccc}\mathbf{1}& \mathbf{1}& \mathbf{3}\\ \mathbf{2}& \mathbf{1}& \mathbf{4}\end{array}\mathbf{\right)}$