Basis

• categories: Linear algebra, Definition

Set of vectors in $V$ that:

1. Spans the vector space: Every vector in $V$ can be expressed as a linear combination of basis vectors.
2. Is linearly independent: No vector in the basis can be written as a linear combination of other basis vectors.

For Matrix = Independent columns that fill Column Space

Intuition

A basis allows each element of $V$ to be uniquely represented by a combination of basis vectors. In finite-dimensional spaces, the basis gives a minimal set of “building blocks” for the space. In infinite-dimensional spaces, a basis may consist of infinitely many vectors but still enables unique representation.