Quotient space

A quotient space V/W consists of equivalence classes of vectors in a vector space V, where vectors are equivalent when their difference lies in a subspace W.

Equivalence relation: A relation that is reflexive, symmetric, and transitive, partitioning a set into equivalence classes. Vectors are grouped precisely when their difference belongs to the chosen subspace.
Vector space: A set with vector addition and scalar multiplication satisfying the vector-space axioms over a field. Both the ambient space and its quotient carry vector-space structure.
Cokernel: The quotient of a linear map's codomain by its image. It measures which codomain directions a linear map fails to reach.
Complementary subspace: A subspace U such that a vector space is the direct sum of U and another subspace W. A complement selects representatives, while the quotient records classes without choosing them.
Coset: A translate of a subgroup by a fixed group element; in vector spaces, a translate of a subspace by a vector. Each element of the quotient is one coset v + W.
Subspace: A subset of a vector space closed under addition and scalar multiplication. The subspace W determines which vectors become equivalent.
Homology group: A quotient of cycles by boundaries in a chain complex, recording holes up to boundary equivalence. Homology uses quotienting to treat cycles that differ by a boundary as equivalent.
Direct sum: A vector space decomposition into subspaces with unique component representations. Direct sums separate directions; quotients identify one subspace's directions with zero.
Kernel of a linear map: The subspace of inputs that a linear map sends to zero. Vectors with the same image differ by an element of the kernel.
Linear combination: A sum of scalar multiples of vectors. Linear combinations of cosets define addition and scalar multiplication in V/W.

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