91影视

Describe the images and kernels of the transformations in Exercisesthrough geometrically.

25. Rotation through an angle$\frac{\mathbf{蟺}}{\mathbf{4}}$ of in the counterclockwise direction (in${\mathbf{鈩�}}^{\mathbf{2}}{\mathbf{鈩�}}^{\mathbf{2}}$).

If a subspace V of$R^3$ contains the standard vectors ${\stackrel{鈫�}{e}}_1,{\stackrel{鈫�}{e}}_2,{\stackrel{鈫�}{e}}_3$then V must be $R^3$.

In Exercises 25through 30, find the matrix B of the linear transformation$\mathit{T}\mathbf{(}\stackrel{\mathbf{鈫�}}{\mathbf{x}}\mathbf{}\mathbf{)}\mathbf{=}\mathit{A}\stackrel{\mathbf{鈫�}}{\mathbf{x}}$ with respect to the basis $\mathbf{貜}\mathbf{=}\left(\stackrel{鈫�}{V_1},--\stackrel{鈫�}{V_m}\right)$ .

$\mathit{A}\mathbf{=}\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\mathbf{;}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}\left[\left.\begin{array}{c}1\\ 1\end{array}\right],{\stackrel{鈫�}{V}}_2=\left|\begin{array}{c}1\\ 2\end{array}\right.\right]$

What is the image of a function ffrom $\mathbf{鈩�}$ to $\mathbf{鈩�}$given by

$\mathit{f}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}{\mathit{t}}^{\mathbf{3}}\mathbf{+}\mathit{a}{\mathit{t}}^{\mathbf{2}}\mathbf{+}\mathit{b}\mathit{t}\mathbf{+}\mathit{c}$,

where a,b,c are arbitrary scalars?

In Exercises 25through 30, find the matrix Bof the linear transformation $\mathit{T}\left(\stackrel{鈫�}{x}\right)\mathbf{=}\mathit{A}\left(\stackrel{鈫�}{x}\right)$ with respect to the basis $\mathfrak{J}\mathbf{=}\left({\stackrel{鈫�}{v}}_1,鈥�..,{\stackrel{鈫�}{v}}_m\right)$.

$\mathit{A}\mathbf{=}\left(\begin{array}{cc}0& 1\\ 2& 3\end{array}\right)\mathbf{;}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{=}\left[\begin{array}{c}1\\ 2\end{array}\right]\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}\mathbf{=}\left[\begin{array}{c}1\\ 1\end{array}\right]$

If a 2 脳 2 matrix P represents the orthogonal projection onto a line in $R^2,$then P must be similar to the matrix$\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$

Consider the matrices

$\begin{array}{l}\mathbf{C}\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{H}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 1\\ 1& 0& 1\end{array}\right]\mathbf{,}\\ \mathbf{L}\mathbf{=}\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 1& 1& 1\end{array}\right]\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{T}\mathbf{=}\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]\mathbf{,}\\ \mathbf{X}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]\mathbf{,}\mathbf{\text{鈥夆赌夆赌�}}\mathbf{Y}\mathbf{=}\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 1& 0\end{array}\right]\end{array}$

1. Which of the matrices in this list have the same kernel as matrix $\mathbf{C}$ ?
2. Which of the matrices in this list have the same image as matrix $\mathbf{C}$?
3. Which of these matrices has an image that is different from the images of all the other matrices in the list?

If\[\overrightarrow v \] and \[\overrightarrow w \] are linearly independent eigenvectors of a symmetric matrix \[A\], then \[\overrightarrow w \] must be orthogonal to \[\overrightarrow v \].

If \(\overrightarrow v \) and \(\overrightarrow w \) are linearly independent eigenvectors of a symmetric matrix \(A\), then \(\overrightarrow w \) must be orthogonal to \(\overrightarrow v \).

Give an example of a noninvertible function Ffrom$\mathbf{鈩�}$to $\mathbf{鈩浓划}$with

$\mathit{i}\mathit{m}\left(f\right)\mathbf{=}\mathbf{鈩�}$