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

  • Context: MHB 
  • Thread starter Thread starter catsarebad
  • Start date Start date
  • Tags Tags
    Matrix
Click For Summary
SUMMARY

The discussion focuses on constructing a matrix A in Mat_n*n (F) such that the cyclic subspace C_x, generated by a vector x, does not span the vector space V. Participants emphasize that choosing a non-invertible matrix A guarantees that C_x will not span V, as demonstrated by the relationship where Ax = 0 for some non-zero x. Consequently, C_x reduces to {x, 0, 0, ...}, which cannot span V. This highlights the importance of matrix properties in linear algebra.

PREREQUISITES
  • Understanding of linear transformations and their representations as matrices.
  • Familiarity with the concept of cyclic subspaces in linear algebra.
  • Knowledge of invertible vs. non-invertible matrices.
  • Basic grasp of vector spaces and spanning sets.
NEXT STEPS
  • Study the properties of non-invertible matrices and their implications in linear transformations.
  • Learn about cyclic subspaces and their generation from vectors in linear algebra.
  • Explore the concept of span and its significance in vector spaces.
  • Investigate the relationship between linear maps and their matrix representations in different bases.
USEFUL FOR

Students and professionals in mathematics, particularly those studying linear algebra, matrix theory, and vector spaces. This discussion is beneficial for anyone looking to deepen their understanding of matrix properties and cyclic subspaces.

catsarebad
Messages
69
Reaction score
0
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),...}
 
Physics news on Phys.org
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 $V$?
 

Similar threads

MHB Show deg of minimal poly = dimension of V
  • · Replies 1 ·
Replies
1
Views
1K
MHB Show all invariant subspaces are of the form
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad If L is diagonalizable, algebraic & geometric multiplicities are equal
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Proof that if T is Hermitian, eigenvectors form an orthonormal basis
  • · Replies 3 ·
Replies
3
Views
4K
MHB Solve Sets of Form $v+X$ in $\mathbb{R}^2$
  • · Replies 5 ·
Replies
5
Views
2K
MHB How Does the Determinant of a Matrix Relate to the Area of a Parallelogram?
  • · Replies 15 ·
Replies
15
Views
2K
MHB Proving or disproving this matrix V is invertible.
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Linear algebra - need to show deg of minimal poly = dimension of V
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Proof by induction of block diagonal decomposition of a matrix
  • · Replies 4 ·
Replies
4
Views
2K