91影视

For the matrix

$\mathit{A}\mathbf{=}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$

Describe the images and kernels of the matrices$\mathit{A}\mathbf{,}\mathbf{鈥�}{\mathit{A}}^{\mathbf{2}}\mathbf{,}\mathbf{鈥�}$ and${\mathit{A}}^3\mathbf{鈥�}$ geometrically.

Consider a square matrix.

1. What is the relationship among ker($\mathit{A}$) and ker (${\mathit{A}}^{\mathbf{2}}$)? Are they necessarily equal? Is one of them necessarily contained in the other? More generally, what can you say about ker($\mathit{A}$),ker (${\mathit{A}}^{\mathbf{2}}$), ker (${\mathit{A}}^{\mathbf{3}}$),鈥�?
2. What can you say about im($\mathit{A}$),im (${\mathit{A}}^{\mathbf{2}}$), im(${\mathit{A}}^{\mathbf{3}}$),鈥�?

Hint: Exercise 37 is helpful.

Consider an $\mathbf{鈥�}\mathit{n}\mathbf{脳}\mathit{p}$matrix $\mathbf{鈥�}\mathit{A}$and$\mathbf{鈥�}\mathit{p}\mathbf{脳}\mathit{m}$matrix $\mathit{B}$.

1. What is the relationship between ker($\mathit{A}\mathit{B}$) and ker role="math" localid="1664188035115" $\mathit{B}$)? Are they always equal? Is one of them always contained in the other?
2. What is the relationship between im($\mathit{A}$) and im (role="math" localid="1664187975738" $\mathit{A}\mathit{B}$)?

Consider an $\mathbf{鈥�}\mathit{n}\mathbf{脳}\mathit{p}$matrix$\mathbf{鈥�}\mathit{A}$and $\mathit{p}\mathbf{脳}\mathit{m}$matrix $\mathit{B}$. If ker ($\mathbf{鈥�}\mathit{A}$) = im ($\mathit{B}$), what can you say about the product $\mathit{A}\mathit{B}$?

Consider a matrix $\mathit{A}\mathbf{=}\left[\begin{array}{cc}0.36& 0.48\\ 0.48& 0.64\end{array}\right]$.

1. Describe ker (role="math" localid="1664194291935" $\mathit{A}$) and im ($\mathit{A}$) geometrically.
2. Find ${\mathit{A}}^{\mathbf{2}}$. If $\mathit{蠀}$is in the image of $\mathit{A}$, what can you say about $\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{蠀}}$?
3. Describe the transformation $\mathit{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$geometrically.

Express the image of the matrix

$\mathit{A}\mathbf{=}\mathbf{\left[}\begin{array}{cccc}\mathbf{1}& \mathbf{1}& \mathbf{1}& \mathbf{6}\\ \mathbf{1}& \mathbf{2}& \mathbf{3}& \mathbf{4}\\ \mathbf{1}& \mathbf{3}& \mathbf{5}& \mathbf{2}\\ \mathbf{1}& \mathbf{4}& \mathbf{7}& \mathbf{0}\end{array}\mathbf{\right]}$

as the kernel of a matrix$\mathit{B}$. Hint: The image of$\mathit{A}$consists of all vectors$\stackrel{\mathbf{鈫�}}{\mathbf{y}}$in${\mathbf{鈩�}}^{\mathbf{4}}$such that the system$\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{=}\stackrel{\mathbf{鈫�}}{\mathbf{y}}$is consistent. Write this system more explicitly:

localid="1664197199135" $\left|\begin{array}{l}{x}_1+鈥�鈥�鈥�鈥�x_2+鈥�鈥�鈥�鈥�x_3+6x_4=y_1\\ x_1+2x_2+3x_3+4x_4=鈥�y_2\\ x_1+3x_2+5x_3+2x_4=鈥�y_3\\ x_1+4x_2+7x_{3鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�}鈥�鈥�=鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�鈥�y_4\end{array}\right|$.

Now, reduce rows:

$\left|\begin{array}{rcl}{\mathit{x}}_{\mathbf{1}}\mathbf{-}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}{\mathit{x}}_{\mathbf{3}}\mathbf{+}\mathbf{8}{\mathit{x}}_{\mathbf{4}}& \mathbf{=}& \mathbf{}\mathbf{}\mathbf{}\mathbf{4}{\mathit{y}}_{\mathbf{3}}\mathbf{-}\mathbf{3}{\mathit{y}}_{\mathbf{4}}\\ {\mathit{x}}_{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathit{x}}_{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathit{x}}_{\mathbf{4}}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{-}{\mathit{y}}_{\mathbf{3}}\mathbf{+}{\mathit{y}}_{\mathbf{4}}\\ \mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{-}\mathbf{3}{\mathit{y}}_{\mathbf{3}}\mathbf{+}\mathbf{2}{\mathit{y}}_{\mathbf{4}}\\ \mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{鈥�}\mathbf{0}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{}\mathbf{鈥�}\mathbf{鈥�}{\mathit{y}}_{\mathbf{2}}\mathbf{-}\mathbf{鈥�}\mathbf{2}{\mathit{y}}_{\mathbf{3}}\mathbf{+}{\mathit{y}}_{\mathbf{4}}\end{array}\right|$

For which the vectors $\stackrel{\mathbf{鈫�}}{\mathbf{y}}$is this system consistent? The answer allows you to express im ( $\mathit{A}$) as the kernel of a$\mathbf{2}\mathbf{脳}\mathbf{4}$matrix $\mathit{B}$.

consider the dynamical system

$\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left[\begin{array}{cc}1.1& 0\\ 0& 位\end{array}\right]\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

Sketch a phase portrait of this system for the given values of:

$R^2$is a subspace of$R^3.$

Give an example of a matrixAsuch thatim(A)is the plane with normal vector $\left[\begin{array}{c}1\\ 3\\ 2\end{array}\right]$in${\mathbf{鈩�}}^{\mathbf{3}}{\mathbf{鈩�}}^{\mathbf{3}}$ .

Let V be the subspace of ${\mathbf{鈩�}}^{\mathbf{4}}$defined by the equation

${\mathbf{x}}_{\mathbf{1}}\mathbf{-}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}\mathbf{4}{\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{0}$

Find a linear transformation T from ${\mathbf{鈩�}}^{\mathbf{4}}$to ${\mathbf{鈩�}}^{\mathbf{4}}$such that $\mathbf{ker}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{=}\left\{\stackrel{鈫�}{0}\right\}$and im(T) = V. Describe T by its matrix A.