Question $17$: $[$Basic$]$ Suppose A is a $3$ by $3$ matrix. Consider a linear system that outputs

y $=$ Ax where x in R

$3$

is the input signal. To be more precise, x is a column vector of

dimensional $3,$ and y in R

$3$

is the output column vector of dimension $3.$ Further assume

that we know that

$$ When x$1=(1,0,0)$T

$,$ the output is y$1=(1,2,3)$T

$.$

$$ When x$2=(0,1,0)$T

$,$ the output is y$2=(3,2,1)$T

$.$

$$ When x$3=(0,0,1)$T

$,$ the output is y$3=(2,1,3)$T

$.$

What is the output y $=$ Ax when the input is x $=(2,2,4)$T

$?($Hint: Use the linearity of

the system.$)$