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construct a matrix with such that V is not equal to C_x

chickenwin

New member
Feb 13, 2014
12
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

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?
 

chickenwin

New member
Feb 13, 2014
12
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

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?
 

chickenwin

New member
Feb 13, 2014
12
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

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
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 $V$?