Is This Proof of Linear Dependence Correct?

[autre]

I have to prove:

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

My attempts at a proof:

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

Am I on the right track?

 

[micromass]

Yes, that's ok.

 

[Deveno]

because...if we write:

v = v₁e₁+v₂e₂+...+v_ne_n,


then: v₁e₁+v₂e₂+...+v_ne_n - 1v

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