91影视

Use Exercise 69 to answer the following question: If$\mathit{V}$ and $\mathit{W}$are subspaces of ${\mathbf{鈩�}}^{\mathbf{10}}$, with $\mathit{d}\mathit{i}\mathit{m}\left(V\right)\mathbf{=}\mathbf{6}$and $\mathit{d}\mathit{i}\mathit{m}\mathbf{\left(}\mathbf{W}\mathbf{\right)}\mathbf{=}\mathbf{7}$, what are the possible dimensions of $\mathit{V}\mathbf{鈭�}\mathit{W}$.

In Exercises 71 through 74, we will study the row space of a matrix. The row space of an $\mathbf{n}脳\mathbf{m}$matrix $\mathbf{A}$is defined as the span of the row vectors of $\mathbf{A}$(i.e., the set of their linear combinations). For example, the row space of the matrix.

$\left[\begin{array}{cccc}1& 2& 3& 4\\ 1& 1& 1& 1\\ 2& 2& 2& 3\end{array}\right]$

Is the set of all row vectors of the form

$\mathbf{a}\left[\begin{array}{cccc}1& 2& 3& 4\end{array}\right]\mathbf{+}\mathbf{b}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{+}\mathbf{c}\mathbf{\left[}\begin{array}{cccc}\mathbf{2}& \mathbf{2}& \mathbf{2}& \mathbf{3}\end{array}\mathbf{\right]}$

Find a basis of the row space of the matrix

role="math" localid="1664291974793" $\mathbf{E}\mathbf{=}\mathbf{\left[}\begin{array}{ccccc}\mathbf{0}& \mathbf{1}& \mathbf{0}& \mathbf{2}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{1}& \mathbf{3}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{1}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{0}\end{array}\mathbf{\right]}$

In Exercises 71 through 74, we will study the row space of a matrix. The row space of an $\mathit{n}\mathbf{脳}\mathit{m}$matrix $\mathit{A}$is defined as the span of the row vectors of $\mathit{A}$(i.e., the set of their linear combinations). For example, the row space of the matrix

$\left[\begin{array}{cccc}1& 2& 3& 4\\ 1& 1& 1& 1\\ 2& 2& 2& 3\end{array}\right]$

Is the set of all row vectors of the form

$\mathit{a}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{2}& \mathbf{3}& \mathbf{4}\end{array}\mathbf{\right]}\mathbf{+}\mathit{b}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{1}\end{array}\mathbf{\right]}\mathbf{+}\mathit{c}\mathbf{\left[}\begin{array}{cccc}\mathbf{2}& \mathbf{2}& \mathbf{2}& \mathbf{3}\end{array}\mathbf{\right]}$

Consider an $\mathit{n}\mathbf{脳}\mathit{m}$matrix$\mathit{E}$in reduced row-echelon form. Using your work in Exercise 71 as a guide, explain how you can find a basis of the row space of $\mathit{E}$. What is the relationship between the dimension of the row space and the rank of $\mathit{E}$?

Consider an arbitrary $\mathit{n}\mathbf{脳}\mathit{m}$matrix $\mathit{A}$.

1. What is the relationship between the row spaces of $\mathit{A}\mathit{a}\mathit{n}\mathit{d}\mathit{E}\mathbf{=}\mathit{r}\mathit{r}\mathit{e}\mathit{f}\mathbf{\left(}\mathit{A}\mathbf{\right)}$? Hint: Examine how the row space is affected by elementary row operations.
2. What is the relationship between the dimension of the row space of$\mathit{A}$and the rank of $\mathit{A}$?

Find a basis of the row space of the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{cccc}1& 1& 1& 1\\ 2& 2& 2& 2\\ 1& 2& 3& 4\\ 1& 3& 5& 7\end{array}\right]$

Consider an $\mathit{n}\mathbf{脳}\mathit{n}$matrix $\mathit{A}$. Show that there exist scalars ${\mathit{c}}_{\mathbf{0}}\mathbf{,}{\mathit{c}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{c}}_{\mathbf{n}}$(not all zero) such that the matrix${\mathit{c}}_{\mathbf{0}}{\mathit{I}}_{\mathbf{n}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathit{A}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{A}}^{\mathbf{2}}\mathbf{+}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathit{A}}^{\mathbf{n}}$is noninvertible. Hint: Pick an arbitrary nonzero vector$\stackrel{\mathbf{鈫�}}{\mathbf{v}}$in ${\mathrm{鈩�}}^n$. Then the$\mathit{n}\mathbf{+}\mathbf{1}$vectors$\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}{\mathbf{A}}^{\mathbf{2}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathbf{A}}^{\mathbf{n}}\stackrel{\mathbf{鈫�}}{\mathbf{v}}$will be linearly dependent. (Much more is true: There are scalars${\mathit{c}}_{\mathbf{0}}\mathbf{,}{\mathit{c}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{c}}_{\mathbf{n}}\mathbf{,}$not all zero, such that${\mathit{c}}_{\mathbf{0}}{\mathit{I}}_{\mathbf{n}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathit{A}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{A}}^{\mathbf{2}}\mathbf{+}\mathbf{路}\mathbf{路}\mathbf{路}\mathbf{+}{\mathit{c}}_{\mathbf{n}}{\mathit{A}}^{\mathbf{n}}\mathbf{=}\mathbf{0}$. You are not asked to demonstrate this fact here.)

Consider the matrix

$\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]$.

Find scalars${\mathit{c}}_{\mathbf{0}}\mathbf{,}{\mathit{c}}_{\mathbf{1}}\mathbf{,}{\mathit{c}}_{\mathbf{2}}$ (not all zero) such that the matrix${\mathit{c}}_{\mathbf{0}}{\mathit{I}}_{\mathbf{2}}\mathbf{+}{\mathit{c}}_{\mathbf{1}}\mathit{A}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{A}}^{\mathbf{2}}$ is noninvertible. See Exercise 75.

Consider an$\mathit{n}\mathbf{脳}\mathit{m}$ matrix . Show that the rank of$\mathit{A}$ is $\mathit{n}$if (and only if)$\mathit{A}$has an invertible $\mathit{n}\mathbf{脳}\mathit{n}$submatrix (i.e., a matrix obtained by deleting$\mathit{m}\mathbf{-}\mathit{n}$columns of $\mathit{A}$.

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 $\mathit{A}$.

role="math" localid="1664273160172" $\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 4\end{array}\right)$

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 $\mathit{A}$.

$\left(\begin{array}{ccc}0& 1& 1\\ 0& 1& 2\\ 0& 1& 3\end{array}\right)$