Question

Suppose A = -3370 , * 30 and f(x) = Ax. a. If possible, complete the following equations; otherwise, enter DNE. [-14 ][-2] - DNE $(1–8, –4) = 2 (-8, 4) f(-8, - 4)) = (5,2) in the subspace b. The linear transformation f acts like multiplication by 2 span{(-2, -1)}. c. Is the vector (-2,-1) an eigenvector for A? yes If so, what is its associated eigen d. Is the vector (-8, 4) an eigenvector for A? yes If so, what is its associated eigen

Answer 1

so most of them is solved

for a

$\begin{bmatrix} -33 & 70\\ -14& 30 \end{bmatrix}$$\begin{bmatrix} -2\\ -1 \end{bmatrix}=$$\begin{bmatrix} -4\\ -2 \end{bmatrix}=$$2\begin{bmatrix} -2\\ -1 \end{bmatrix}$

$\begin{bmatrix} -33 & 70\\ -14& 30 \end{bmatrix}$$\begin{bmatrix} -8\\ -4 \end{bmatrix}=$$\begin{bmatrix} -16\\ -8\end{bmatrix} \neq k\begin{bmatrix} 5\\2 \end{bmatrix} for \: any \: k$

for g.

we need to find (x,y) such that

$\begin{bmatrix} -33 & 70\\ -14& 30 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}=-5 \begin{bmatrix} x\\ y \end{bmatrix}$

i.e. -33x+70y=-5x and -14x+30y=-5y

i.e. -28x+70y=0 and -14x+35y=0

i.e. 2x=5y

so our vector is of the form (5,2)x.so answer is <5,2>.

h.

$\begin{bmatrix} -33 & 70\\ -14& 30 \end{bmatrix}\begin{bmatrix} -15\\ -6 \end{bmatrix}=\begin{bmatrix} -75\\ -30 \end{bmatrix}$$= -5\begin{bmatrix} -15\\ -6 \end{bmatrix}$

so eigen value is -5.