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  1. MHB Apprentice

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    #1
    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),.......}

  2. MHB Master
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    #2
    This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?

  3. MHB Apprentice

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    Quote Originally Posted by Deveno View Post
    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

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

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    #4
    Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?

  5. MHB Apprentice

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    Quote Originally Posted by Deveno View Post
    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?

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    #6
    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 $V$?

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