for each n greater than or equal to,
construct a matrix A that belongs to Mat_n*n (F) such that
V is not equal to C_x for every x that belongs to V
here,
C_x = span {x, L(x), L^2(x), .............L^k(x),.......}
for each n greater than or equal to,
construct a matrix A that belongs to Mat_n*n (F) such that
V is not equal to C_x for every x that belongs to V
here,
C_x = span {x, L(x), L^2(x), .............L^k(x),.......}
This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?
Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?
If $Ax = 0$ (which is true for SOME non-zero $x$ if $A$ is not invertible) then:
$C_x = \{x,Ax,A^2x,\dots\} = \{x,0,0,\dots\}$
HOW CAN THIS SPAN $V$?