Are Linear Transformations of Linearly Dependent Sets Also Linearly Dependent?

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Discussion Overview

The discussion centers on the properties of linear transformations, specifically whether linear transformations of linearly dependent sets of vectors remain linearly dependent. It also explores the implications for linearly independent sets when transformed by a matrix, particularly focusing on the role of invertibility.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant proposes that if {v1, v2, v3} is a linearly dependent set in R^3, then {Av1, Av2, Av3} is also linearly dependent, citing the preservation of vector operations by linear transformations.
  • Another participant provides a mathematical argument showing that if the original vectors are linearly dependent, then the transformed vectors must also satisfy a linear dependence relation.
  • A participant expresses understanding of the previous points and raises a related question about the effect of a linear transformation on a linearly independent set, particularly questioning the impact of the matrix being invertible.
  • Another participant discusses the contrapositive of the initial claim, suggesting that if the transformed vectors are linearly independent, then the original vectors must also be independent, but notes that this does not hold for arbitrary matrices.
  • The same participant concludes that if the transformation matrix is invertible, then the independence of the original vectors is preserved in the transformed set.

Areas of Agreement / Disagreement

Participants generally agree on the implications of linear transformations for linearly dependent sets, but there is ongoing exploration regarding the effects on linearly independent sets, particularly concerning the conditions of invertibility. The discussion remains unresolved regarding the broader implications for arbitrary matrices.

Contextual Notes

Limitations include the assumption that the matrix A is not the zero matrix when discussing linear independence, and the need for further exploration of conditions under which linear independence is preserved.

Mola
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If A is a 3x3 Matrix and {v1, v2, v3} is a linearly dependent set of vectors in R^3, then {Av1, Av2, Av3} is also a linearly dependent set?

Is this true? Can someone please explain why or why not??

What I think: I think it is true because I read that a linear transformation preserves the operations of vector addition and scalar multiplication.
 
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If v1,v2,v3 are linearly dependent you can find constants a1, a2, a3 not all 0 such that
a_1v_1 + a_2v_2 + a_3v_3 = 0
Now left-multiplying this by A you get:
A(a_1v_1+a_2v_2+a_3v_3) = A0=0
Now use your rules for matrix arithmetic to derive:
a_1(Av_1)+a_2(Av_2)+a_3(Av_3)=0
(HINT: Ak = kA for constants k, and A(v+w) = Av+Aw for vectors v, w where the expression makes sense).
 
That makes sense. So if we have a1(Av1) + a2(Av2) + a3(Av3) = 0, then at least one of the constants could be zero and that will definitely result to a linearly dependent set.
Thanks.

That leads me to a related theory: Let's assume we are talking about {v1, v2, v3} being a linearly INDEPENDENT set now. If we multiply the vectors by the matrix A, how does it affect the independece? Would it make a differerence if the matrix A is invertible?
 
Mola said:
That leads me to a related theory: Let's assume we are talking about {v1, v2, v3} being a linearly INDEPENDENT set now. If we multiply the vectors by the matrix A, how does it affect the independece? Would it make a differerence if the matrix A is invertible?

This is actually a quite interesting little question (well in my opinion anyway). First for fixed A, v1,v2,v3 note that if we take the contrapositive of your initial result we get:
If Av1, Av2, Av3 are linearly independent, then v1,v2,v3 are linearly independent.
so for linear independence it goes backwards. For an arbitrary matrix A we can not prove your new statement since we can just let A be the 0 matrix. However if A is invertible, then we can just go backwards by noting that if,
a_1Av_1+a_2Av_2+a_3Av_3 = 0
Then we can left-multiply by A^{-1} to get,
a_1v_1+a_2v_2+a_3v_3 = 0
so if v1,v2,v3 are linearly independent and A is invertible, then Av1, Av2, Av3 are linearly independent.
 
Thanks rasmhop... I did think "A" being an invertible matrix could make a difference but I didn't know how to prove it.
That was a very good help from you.
 

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