Two theorems regarding bases of vector spaces... how are they different?

Question

Assume that $V$ can be generated by $n$ vectors. Then any sequence of vectors of length greater than $n$ is linearly dependent.

$\vdots$

Every spanning list in a vector space can be reduced to a basis of the vector space.

The first theorem claims that if we have a sequence $S = (v_1,\ldots,v_n)$ of $n$ vectors such that $\operatorname{span}(S) = V,$ then adding more vectors to $S$ makes it linearly dependent in $V$ (and so not a basis), while the second theorem states that if we already have a sequence $S$ such that $\operatorname{span}(S) = V,$ then we can remove vectors to make $S$ a basis (In particular, linearly independent). So in summary, we can add vectors to a spanning list and it won't be a basis, but we can take away vectors until the spanning list is a basis...

Question: It appears that either one of these theorems can be used interchangeably to prove that every finite-dimensional vector space has a basis, so how are the two theorems different? They seem so similar to me...

Answer 1

The theorems are essentially discussing the nature of a basis. A basis for a space must be both spanning and linearly independent. So, a basis $B$ for a vector space $V$, must have at least $dim V$ elements, and at most $dim V$ elements. The dimension of $V$ is exactly the dimension of any of its bases.

That is, the first statement gives you an upper bound on the size of a basis, and the second gives you a lower bound on basis length.