$6\mathrm{.}8($Discrete$-$time systems$)$ Consider a linear discrete$-$time system of the form x$[$k $+1]=$ Ax$[$k$]+$ Bu$[$k$],$ y$[$k$]=$ Cx$[$k$]+$ Du$[$k$].($a$)$ Show that the general form of the output of a discrete$-$time linear system is given by the discrete$-$time convolution equation: y$[$k$]=$ CAkx$[0]+$ k $\%1$ j$=0$ CAk$$j$1$Bu$[$j$]+$ Du$[$k$].($b$)$ Show that a discrete$-$time linear system is asymptotically stable if and only if all the eigenvalues of A have a magnitude strictly less than $1.($c$)$ Show that a discrete$-$time linear system is unstable if any of the eigenvalues of A have magnitude greater than $1.$