My Math Forum need help understanding proof

 Linear Algebra Linear Algebra Math Forum

 October 18th, 2017, 11:24 AM #1 Member   Joined: Jan 2016 From: Blackpool Posts: 96 Thanks: 2 need help understanding proof Let v1,...,vn be vectors in Rn. Then {v1,...,vn} spans Rn if and only if, for the matrix A = v1 v2 ··· vn , the linear system Ax = v is consistent for every v ∈ Rn. I understand the proof but I'm unsure with how the word "consistent" is used, does it mean v should be the same for each vector? Thanks!
 October 18th, 2017, 01:17 PM #2 Global Moderator   Joined: May 2007 Posts: 6,556 Thanks: 600 I suspect it means there is a unique x for each v.
October 18th, 2017, 06:28 PM   #3
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 Originally Posted by mathman I suspect it means there is a unique x for each v.
There is no need to impose uniqueness. Consistent means simply that the equation has a solution. Of course, as a consequence you can additionally prove uniqueness but this is not necessary to prove directly nor does the definition of consistency require it.

The main idea is that if there exists some $v$ such that $Ax = v$ can't be solved, then this is equivalent to saying that $v$ is not in the image of $A$. This of course is equivalent to saying that $v$ is not in the span of the columns of $A$ implying that $A$ is not full rank.

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