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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?