The dimension of the kernel (null space) of a linear transformation between finite-dimensional vector spaces; equivalently the number of linearly independent solutions to the homogeneous equation T(v)=0.

Principle

Rank–nullity principle: for a linear map T: V → W between finite-dimensional spaces, dim V = rank(T) + nullity(T). Nullity measures the degrees of freedom lost to mapping into zero.

Demonstration

Example: a linear map T: R^4 → R^3 whose matrix in a standard basis has rank 2 has nullity 4−2=2; there are two independent free variables in the homogeneous system T(v)=0, so the kernel is a 2-dimensional subspace of R^4.

Misapplication

Treating nullity as the algebraic multiplicity of the eigenvalue 0 without checking that eigenvectors span a kernel, or applying rank–nullity unchanged in infinite-dimensional settings where dimensions can be infinite and subtleties of topological closure occur.

Consequence

Correct use of nullity identifies the dimension of the solution space of homogeneous linear systems, informs count of free parameters in parametric solutions, and determines noninjectivity: nullity>0 implies T is not injective.

Reversal

Invert the perspective to consider codimension of the image (corank): instead of counting kernel dimensions, count how many dimensions of the codomain are missed by the image.

Boundary

Applies to linear maps between finite-dimensional vector spaces over a field; for infinite-dimensional spaces, or for nonlinear maps, analogous notions require additional structure (topology, Fredholm index, etc.) and may not reduce to a finite integer.

Semantic Tension

Nullity versus algebraic multiplicity of zero eigenvalue: nullity equals geometric multiplicity but can be less than algebraic multiplicity; also tension between nullity and notions like index or cokernel in dual settings.

Synthesis

Nullity is the integer dimension of the kernel of a linear operator; by the rank–nullity relation it complements rank and precisely counts independent homogeneous solutions.