Assume that a spanning set $S$ for a vector space of dimension $n$ is not linearly independent.
Then we can write at least one of the vectors in $S$ as a linear combination of the others:
Now, if $S$ is a spanning set, every element of $u\in V$ can be written as a linear combination of elements from $S$
$$u=\sum_i a_i v_i,$$
but rewriting $v_j$ in terms of the other $v_i$ shows that every vector $u\in V$ can be written as a linear combination of $n-1$ vectors:
Hence there is a basis of $V$ size less than or equal to $n-1$. This contradicts $\dim V=n$ and so this spanning set must be linearly independent.