Basis and span
The span of vectors contains all their linear combinations. A basis is an independent set that spans a vector space.
Definition
The span of vectors contains all their linear combinations. A basis is an independent set that spans a vector space.
Intuition
A basis is a complete coordinate system without redundant directions.
Worked example
(1, 0) and (0, 1) form a basis of the plane. The single vector (1, 1) spans only a line.
The math
Every vector has a unique linear combination in a chosen basis. The number of basis vectors is the dimension.
In machine learning
Choose a basis to simplify a transformation, compress a representation, or interpret feature directions.
Go deeper
Sources
More in Linear algebra
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