MHB Construct a matrix with such that V is not equal to C_x

[catsarebad]

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),...}

 

[Deveno]

This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?

 

[catsarebad]

This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?

I do not know (Speechless)

I cannot figure out how A correlated to C_x. Could you please explain that to me? Thanks a ton. (heart)

 

[Deveno]

Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?

 

[catsarebad]

Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?

still dunno. i think i just do not get what A is with respect to C_x. i know C_x is cyclic subspace generated by x that is spanned by vectors, x, L(x),...

but what is A? how does it relate to x, L(x), C_x, etc?

 

[Deveno]

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 $V$?