Dimension theorem for vector spaces

For a linear map between finite-dimensional vector spaces, the dimension of the domain equals the dimension of its kernel plus the dimension of its image.

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Kernel (linear algebra): The subspace of a linear map’s domain consisting of vectors mapped to zero. Its dimension is the part of the domain that the theorem separates from the image.
Linear map: A function between vector spaces that preserves vector addition and scalar multiplication. The theorem applies to this structure-preserving function.
System of linear equations: A collection of linear equations sharing the same unknowns. The theorem relates the number of free solution directions to the number of independent constraints.
Infinite-dimensional vector space: A vector space that has no finite basis. Finite-dimensionality is essential to the theorem’s ordinary finite dimension equation.
Image (linear algebra): The subspace of a linear map’s codomain consisting of all outputs the map can produce. Its dimension is the portion of the domain’s dimension accounted for by independent outputs.
Linear independence: A property of vectors where only the trivial linear combination equals zero. Independent vectors in the kernel remain part of a basis when extended in the domain.
Homogeneous system of linear equations: A system of linear equations whose constant terms are all zero. Its solution space is the kernel whose dimension the theorem counts.
Hilbert hotel paradox: A thought experiment showing that an infinite set can be put in bijection with a proper subset. Infinite-dimensional spaces can have the same dimension despite a nontrivial kernel.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. A kernel basis extended to a domain basis yields the theorem’s dimension count.
Span (linear algebra): The set of all linear combinations of a given collection of vectors. Basis extension relies on adding vectors until the collection spans the domain.