Proving linear dependence (2.0)

[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).