Linear Dependence in Rn with Nonsingular Matrix A

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Homework Help Overview

The problem involves linear dependence of vectors in Rn, specifically examining the relationship between linearly dependent vectors x1, x2, x3 and their transformations y1, y2, y3 through a nonsingular matrix A.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of the linear dependence of the original vectors and question the necessity of the zero vector in the solution. There are inquiries about the assumptions regarding the nonsingularity of matrix A and its relevance to the problem.

Discussion Status

The discussion is ongoing with participants questioning each other's interpretations and clarifying the conditions of the problem. Some guidance has been offered regarding the implications of the assumptions, but no consensus has been reached.

Contextual Notes

There is a noted uncertainty regarding the necessity of the matrix A being nonsingular and its impact on the linear dependence of the transformed vectors.

aargoo
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Homework Statement


Let x1,x2,x3 be linearly dependent vectors in Rn, let A be a nonsingular n x n matrix, and let y1=Ax1, y2=Ax2, y3=Ax3. Prove that y1, y2,y3 are linearly dependent.




Homework Equations





The Attempt at a Solution


My solution was y is equal to the zero vector, thus must be linearly dependent.
 
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What is y in your solution? You have specified y1, y2, and y3 ...
 
I think you must have misunderstood the problem. y certainly does NOT have to be the 0 vector.
 
What does the assumption tell you about the vectors ##x_1,x_2,x_3##? The answer to the problem follows almost immediately from the answer to this question.
 
Without trying to derail the thread too much, I don't see why ##A## has to be nonsingular.
 

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