Invertability as a substitute for looking for linear independence

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Discussion Overview

The discussion revolves around the relationship between the invertibility of a matrix and the linear independence of a set of vectors represented by its columns. Participants explore whether invertibility can serve as a substitute for determining linear independence, particularly in the context of square and non-square matrices.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant asks if invertibility can be used as an alternative method to determine linear independence.
  • Another participant asserts that the columns of a matrix are linearly independent if and only if the matrix is invertible, specifically for square matrices.
  • A further contribution states that if a set creates a matrix A, then the set is linearly independent if the equation Ax = 0 has only the trivial solution, linking this to the invertibility of A.
  • Another participant reiterates the previous point about linear independence and invertibility for square matrices but notes that non-square matrices, while not invertible, can still consist of linearly independent vectors.

Areas of Agreement / Disagreement

Participants generally agree that for square matrices, invertibility is linked to linear independence. However, there is a recognition that non-square matrices complicate this relationship, indicating some disagreement on the applicability of the concept across different types of matrices.

Contextual Notes

The discussion does not resolve the implications of non-square matrices on linear independence and does not clarify the conditions under which the statements apply.

ichigo444
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Can i use the invertability of a matrix as an alternative way of determining the linear independence of a set? Thank you.
 
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Yes. If the columns of the matrix are considered to be vectors, these vectors are linearly independent iff the matrix is invertible.
 
If you suppose the set creates a matrix A, then you can say the set is linearly independent if Ax = 0 has only the trivial solution. If A is invertible then x = A-1 0 = 0 is the only solution and the set is linearly independent.
 
Mark44 said:
Yes. If the columns of the matrix are considered to be vectors, these vectors are linearly independent iff the matrix is invertible.

This is true for a square matrix. A non-square matrix is obviously not invertible but can be made of linearly independent vectors.
 

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