## Independence-Dimension Inequality

This inequality states that if the $$n$$-vectors $$a_1, \cdots, a_k$$ are linearly independent, then $$k \leq n$$. Meaning that a linearly independent collection of $$n$$-vectors can have at most $$n$$ elements. The concept of a basis relies on this inequality.

## Basis

A collection of $$n$$ linearly independent $$n$$-vectors is called a basis. If the $$n$$-vectors $$a_n, \cdots, a_n$$ are a basis, then any $$n$$-vector $$b$$ can be written as a linear combination of them. Essentially a basis is a set of linearly independent vectors that span a vector space (a collection of vectors).

When we express the $$n$$-vector $$b$$ as a linear combination of a basis $$a_1, \cdots, a_n$$, we refer to

$b = \alpha_1 a_1 + \cdots + \alpha_n a_n$

as the expansion of $$b$$ in the $$a_1, \cdots, a_n$$ basis. The numbers $$\alpha_1, \cdots, \alpha_n$$ are the coefficients of the expansion of $$b$$ in the basis $$a_1, \cdots, a_n$$.