# Proving linear dependence (2.0)

## Main Question or Discussion Point

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?

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