Proving linear dependence (2.0)

  • Thread starter autre
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  • #1
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Main Question or Discussion Point

I have to prove:

Consider [itex]V=F^{n}[/itex]. Let [itex]\mathbf{v}\in V/\{e_{1},e_{2},...,e_{n}\}[/itex]. Prove [itex]\{e_{1},e_{2},...,e_{n},\mathbf{v}\}[/itex] is a linearly dependent set.

My attempts at a proof:

Since [itex]{e_{1},e_{2},...,e_{n}}[/itex] is a basis, it is a linearly independent spanning set. Therefore, any vector [itex]\mathbf{v}\in V[/itex] can be written as a linear combination of [itex]{e_{1},e_{2},...,e_{n}}[/itex]. Therefore, the set [itex]\{e_{1},e_{2},...,e_{n},\mathbf{v}\}[/itex] with [itex]\mathbf{v}\in V[/itex] must be linearly dependent.

Am I on the right track?
 

Answers and Replies

  • #2
22,097
3,279
Yes, that's ok.
 
  • #3
Deveno
Science Advisor
906
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because...if we write:

v = v1e1+v2e2+...+vnen,


then: v1e1+v2e2+...+vnen - 1v

is a linear combination of {e1,e2,...,en,v} that sums to the 0-vector, and yet not all the coefficients in this sum are 0 (the one for v is -1).
 

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