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 \(a_1, \cdots, a_n\).