Answer
(a) $v =\lt x, 2x-5,x-1 \gt$
(b) No possible solutions
Work Step by Step
(a) Let $v= \lt x,y,z \gt$
then
$\lt 1,2,1 \gt \times v=\lt 1,2,1 \gt \times \lt x,y,z \gt= \lt 2z-y, x-z, y-2x \gt$
and
$ \lt 2z-y, x-z, y-2x \gt = \lt 3,1,-5 \gt$
which gives us a system of equations.
$2z-y=3$
$x-z=1$
$y-2x =-5$
Simplify it.
$z=x-1$, $2x-y=5$ or $y=2x-5$
Hence, $v =\lt x, 2x-5,x-1 \gt$
(b) Let $v= \lt x,y,z \gt$
then
$\lt 1,2,1 \gt \times v=\lt 1,2,1 \gt \times \lt x,y,z \gt= \lt 2z-y, x-z, y-2x \gt$
and
$ \lt 2z-y, x-z, y-2x \gt = \lt 3,1,-5 \gt$
which gives us a system of equations.
$2z-y=3$ ... (1)
$x-z=1$ ... (2)
$y-2x =-5$ ...(3)
Simplify it.
$z=x-1$ .... (4)
, $2x-y=5$ or $y=2x-5$ .... (5)
If we add equations (3) and (5), we get $0=10$ which means the system is inconsistent and there are no solutions.
