# Linear algebra proof on linearly independence

## Homework Statement

Show that if vectors v1 , . . . , vk in a vector space V have the properties that v1
does not = 0, and each vi is not in the span of the preceding ones, then the vectors are linearly independent.
Conversely, show that if v1 , . . . , vk is an ordered list of linearly independent vectors, then it has the above properties.

## The Attempt at a Solution

I know it is kinda easy to prove the set is linearly independent, but with the property there, how would I start the proof? I think since v2 is not a multiple of v1, and v3 is not a multiple of vi and v2 ... as well as v1 does not = 0. so i need to show: av1+bv2+cv3......nvn=0 becuz v1 does not = 0, and vi is not the span of the preceding ones(so they are all not =0), does it mean a=b=c...=0?

thanks

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Dick
Homework Helper
You would write a formal proof using induction. There's not much to prove for the k=1 case. Now assume it's true for k and show it's true for k+1. How would that proof look?

is proof by induction the only way to do it? my prof has not really taught how to prove by induction

Dick