Linear Independence

Definition: Linear Independence

Let ${\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n$ be vectors in some vector space $(V,F,+,\cdot$).

We say that ${\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n$ are linearly independent iff

$$c_1{\mathbf{v}}_1+\cdots +c_n{\mathbf{v}}_n=0⟹c_1=\cdots =c_n=0$$

Definition: Maximality

A set of linearly independent vectors $S=\{{\mathbf{v}}_1,{\mathbf{v}}_2,\cdots \}$ is maximal if there is no $\mathbf{v}\in V\setminus S$ such that $S\cup \{\mathbf{v}\}$ are still linearly independent.

Theorem: Size Limit for Linearly Independent Sets

The number of elements in any set $I$ of linearly independent vectors from a finitely generated vector space $(V,F,+,\cdot )$ is always less than or equal to the dimension of $V$.

$$|I|\le \dim (V)$$

PROOF

Let $B=\{{\mathbf{b}}_1,\cdots ,{\mathbf{b}}_n,\}$ be a basis of $V$ and let $I=\{{\mathbf{v}}_1,\cdots ,{\mathbf{v}}_m\}$ be a set of linearly independent vectors.

According to the Steinitz exchange lemma there are $n-m$ vectors in $B$ which form a basis with the vectors from $I$. This means that $n-m$ cannot be negative and thus the proof is complete.