Linear algebra help: Linear independence

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Homework Statement


Let A be an m x n matrix of rank n. Suppose [tex]v_1, v_2, ..., v_k \in \mathbb{R}^n[/tex] and [tex]\{v_1, v_2, ..., v_k\}[/tex] is linearly independent. Prove that [tex]\{Av_1, Av_2, ..., Av_k\}[/tex] is likewise linearly independent.



Homework Equations





The Attempt at a Solution


It says I need to use rank(A) = n, but I'm not sure how to use that info.
 

Answers and Replies

  • #2
Dick
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Rank n tells you that if you have a basis [itex]e_1, e_2, ..., e_n[/itex] of [itex]\mathbb{R}^n[/itex] then [itex]A(e_1), A(e_2), ..., A(e_n)[/itex] are linearly independent. I.e. A takes R^n into an n-dimensional subspace of R^m.
 
  • #3
Deveno
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have you proven a subset of a linearly independent set is also linearly independent?

if so, extend the v's to a basis.
 

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