# Proving that linear independence, spans and basis are equivalent affirmations of a Vectorial Space

Be $X$ a vectorial space of size $n$, $n \ge 1$ and $u_1,u_2,\ ... \ u_n$ vectors from $X$.

Prove that these affirmations are equivalent:

1. $(u_1,u_2...u_n)$ is linear independent;
2. $(u_1,u_2...u_n)$ spans $X$;
3. $\{u_1,u_2...u_n\}$ is a basis of $X$.

Of course, 3. $\implies$ 1. and 3. $\implies$ 2. by the definition of basis.
If 1. holds, then $\langle u_1,\ldots,u_n\rangle$ has dimension $n$. But $\dim X=n$. Therefore $\langle u_1,\ldots,u_n\rangle=X$ and so $\{u_1,\ldots,u_N\}$ is a basis.
If 2. holds, suppose that $\{u_1,\ldots,u_n\}$ is linearly dependent. But then one of the vectors is a linear combination of all others, and so you can forget it. So, $\dim\langle u_1,\ldots,u_n\rangle<n$, which is impossible. Therefore, $\{u_1,\ldots,u_n\}$ is linearly dependent and so it's a basis.