Is This Proof of Linear Dependence Correct?

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SUMMARY

The proof of linear dependence for the set {e1, e2, ..., en, v} is established by recognizing that {e1, e2, ..., en} forms a basis in the vector space V=Fn. Since any vector v in V can be expressed as a linear combination of the basis vectors, the inclusion of v in the set leads to a linear dependence. Specifically, the expression v = v1e1 + v2e2 + ... + vnen demonstrates that the coefficients cannot all be zero, confirming the dependence.

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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?
 
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Yes, that's ok.
 
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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