# Proving linear dependence (2.0)

1. Nov 17, 2011

### 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?

2. Nov 17, 2011

### micromass

Yes, that's ok.

3. Nov 17, 2011

### Deveno

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).