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

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Discussion Overview

The discussion revolves around constructing a matrix \( A \) in the context of linear algebra, specifically focusing on the relationship between the matrix \( A \), a vector \( x \), and the cyclic subspace \( C_x \) generated by \( x \). Participants explore conditions under which \( C_x \) does not span a vector space \( V \), particularly when \( A \) is not invertible.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that choosing a "bad" matrix (non-invertible) will ensure that \( C_x \) does not span \( V \).
  • One participant expresses confusion about how matrix \( A \) relates to \( C_x \) and seeks clarification on this relationship.
  • Another participant points out that if \( Ax = 0 \) for some non-zero \( x \), then \( C_x \) simplifies to \( \{x, 0, 0, \dots\} \), questioning how this can span \( V \).

Areas of Agreement / Disagreement

Participants do not reach a consensus; there are competing views on the implications of choosing a non-invertible matrix and the relationship between \( A \) and \( C_x \). Some participants appear to agree on the idea that non-invertibility affects spanning, while others express uncertainty and seek further clarification.

Contextual Notes

Participants highlight limitations in understanding the correlation between matrix \( A \) and the cyclic subspace \( C_x \), particularly regarding the implications of non-invertibility and the nature of the vectors involved.

Who May Find This Useful

This discussion may be useful for students or individuals studying linear algebra, particularly those interested in the properties of matrices and their effects on vector spaces.

catsarebad
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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),...}
 
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This is easy. Hint: pick a "bad" matrix (one that is not invertible). Why will this guarantee that $C_x$ will not span $V$?
 
Deveno said:
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)
 
Suppose $A$ is such that $Ax = 0$ for some non-zero $x$. What can you say about $C_x$ then?
 
Deveno said:
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?
 
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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