Finite-dimensional vector space

A vector space with a finite basis. The number of vectors in any basis is its dimension, which counts its independent directions.

Vector space: A set with vector addition and scalar multiplication satisfying axioms that include associativity, distributivity, and additive inverses. Finite-dimensional vector spaces are the vector spaces whose bases contain finitely many vectors.
Coordinate vector: A coordinate vector is a list of scalars representing a vector relative to an ordered basis. Choosing a basis turns each vector into a finite tuple of scalars.
Matrix: A matrix is a rectangular array of scalars used to represent linear maps and organize linear equations. Matrices encode linear maps after bases are chosen for finite-dimensional spaces.
Infinite-dimensional vector space: A vector space is infinite-dimensional when it has no finite basis. It lacks the finite coordinate description that defines this concept.
Linear combination: An expression formed by multiplying vectors by scalars and adding the results. A basis spans the space when every vector is a linear combination of its vectors.
Change of basis: A change of basis converts vector coordinates from one basis to another. Different bases describe the same finite-dimensional vectors with different coordinates.
System of linear equations: A system of linear equations is a collection of equations sharing unknown variables. Its solution set can be analyzed using finite-dimensional linear algebra.
Hilbert space: A Hilbert space is a complete inner-product space that may have finite or infinite dimension. Infinite-dimensional Hilbert spaces extend geometric tools beyond finite-dimensional settings.
Linear independence: A set of vectors is linearly independent when only the trivial linear combination equals zero. A basis must be independent, so none of its vectors is redundant.
Linear transformation: A linear transformation preserves vector addition and scalar multiplication. Finite bases let linear transformations be represented by matrices.

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