91影视

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{7}}\left[\begin{array}{ccc}6& -2& -3\\ -2& 3& -6\\ -3& -6& -2\end{array}\right]$

Representing the reflection about a plane E.

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{14}}\left[\begin{array}{ccc}13& -2& -3\\ -2& 10& -6\\ -3& -6& 5\end{array}\right]$

representing the orthogonal projection onto a plane E.

Consider the linear transformation T ( f ) = f from C鈭� to$\frac{\mathbf{-}\mathbf{b}\mathbf{卤}\sqrt{{\mathbf{b}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{ac}}}{\mathbf{2}\mathbf{a}}$C鈭�. For each of the following eigenvalues, find a basis of the associated eigenspace. See Exercise 62. a. 位 = 1 b. 位 = 0 c. 位 = 鈭�1 d. 位 = 鈭�4.

find an eigenbasis for the given matrice and diagonalize:

$\mathit{A}\mathbf{=}\frac{\mathbf{1}}{\mathbf{14}}\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$

In all parts of this problem, let V be the linear space of all 2 脳 2 matrices for which $\left[\begin{array}{c}1\\ 2\end{array}\right]$is an eigenvector.

(a) Find a basis of V and thus determine the dimension of V.

(b) Consider the linear transformation T (A) = A$\left[\begin{array}{c}1\\ 2\end{array}\right]$ from V to ${\mathbf{R}}^{\mathbf{2}}$. Find a basis of the image of Tand a basis of the kernel of T. Determine the rank of T .

(c) Consider the linear transformation L(A) = A$\mathbf{\left[}\begin{array}{c}\mathbf{1}\\ \mathbf{3}\end{array}\mathbf{\right]}$ from V to ${\mathbf{R}}^{\mathbf{2}}$. Find a basis of the image of L and a basis of the kernel of L. Determine the rank of L.

(a) Give an example of a 3 脳 3 matrix A with as many nonzero entries as possible such that both span(${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{1}}$) and span(${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{1}}$,${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{2}}$) are A-invariant subspaces of . See Exercise 65.

(b) Consider the linear space Vof all 3 脳 3 matrices A such that both span (${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{1}}$) and span (${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{1}}$,${\stackrel{\mathbf{鈫�}}{\mathbf{e}}}_{\mathbf{2}}$) are A-invariant subspaces of ${\mathbf{R}}^{\mathbf{3}}$ . Describe the space V (the matrices in V 鈥渉ave a name鈥�), and determine the dimension of V.

Consider the coyotes鈥搑oadrunner system discussed in Example 7. Find closed formulas for c(t) and r(t), for the initial populations ${\mathit{c}}_{\mathbf{0}}$= 100, ${\mathbf{r}}_{\mathbf{0}}$ = 800.

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

h(t + 1) = 4h(t)-2f(t)

f(t + 1) = h(t) + f(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f(t).

$$\begin{gathered} \mathbf{\left(}\mathit{a}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{100} \\ \mathbf{\left(}\mathit{b}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{200}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{100} \\ \mathbf{\left(}\mathit{c}\mathbf{\right)}\mathit{h}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{600}\mathbf{,}\mathit{f}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{500} \end{gathered}$$

68. If A is an n 脳 n matrix with n distinct eigenvalues 位1,...,位n, what is the dimension of the linear space of all n 脳n matrices S such that AS = SB, where B is the diagonal matrix with the diagonal entries 位1,...,位n? Use exercises 64 and 65 as a guide.

Two interacting populations of coyotes and roadrunners can be modeled by the recursive equations

c(t + 1) = 0.75r(t)

r(t + 1) = 鈭�1.5c(t) + 2.25r(t).

For each of the initial populations given in parts (a) through (c), find closed formulas for c(t) and r(t).

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{c}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{100}\mathbf{,}\mathbf{r}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{200} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{c}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{r}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{100} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{c}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{500}\mathbf{,}\mathbf{r}\mathbf{(}\mathbf{}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{700} \end{gathered}$$