Theorem:
If a set of n vectors spans an n-dimensional vector space, then the set is a basis for that vector space.
Attempt:
Let S be a set of n vectors spanning an n-dimensional vector space. This implies that any vector in the vector space $\left(V, R^{n}\right)$ is a linear combination of vectors in the set S.
It suffice to show that S is linearly independent.
Suppose that S is linearly dependent. Then for some vectors $\vec{v}_{i}, \exists i \in \mathbb{Z}_{1}^{n}$ in S that may be expressed as a linear combination of some vectors in S, the removal of $\vec{v}_{i}$ does not affect the span of set S.
Any hints to bring me forward is highly appreciated.
Thanks in advance.