1 Statement of the theorem
The rank–nullity theorem is a basic identity in linear algebra that links the size of a vector space to the behavior of a linear map defined on it. It says that, for a linear transformation with finite-dimensional domain, the dimension of the domain is split into two parts: the rank, which measures the dimension of the image, and the nullity, which measures the dimension of the kernel.
A common way to express the result is:
\[ \dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T), \]
where \(T:V\to W\) is a linear map between finite-dimensional vector spaces.
1.1 Linear transformations
For a linear transformation \(T:V\to W\), the theorem applies to the domain \(V\). The map sends vectors in \(V\) to vectors in \(W\), and its structure is controlled by two subspaces: the kernel, consisting of vectors mapped to zero, and the image, consisting of all vectors that are actually reached in the codomain.
The theorem shows that these two features account for every dimension in the domain. Vectors in the kernel are “lost” under the map, while vectors outside the kernel contribute to the image.
1.2 Matrix form
When a linear transformation is represented by a matrix \(A\), the theorem becomes a statement about the columns and solutions of the homogeneous system \(Ax=0\). In this setting, the rank is the dimension of the column space of \(A\), and the nullity is the dimension of the solution space of the homogeneous equation.
If \(A\) is an \(m\times n\) matrix, then
\[ n=\operatorname{rank}(A)+\operatorname{nullity}(A). \]
This form is especially useful in computations, since rank can be found from row reduction and nullity from the number of free variables.
1.3 Finite-dimensional vector spaces
The theorem is stated for finite-dimensional vector spaces, where dimensions are ordinary integers. Finiteness is essential in the usual form, because the proof relies on choosing bases and counting basis elements.
In finite dimensions, the theorem provides a complete numerical relation among three quantities: the size of the domain, the dimension of the image, and the dimension of the kernel.
1.4 Rank and nullity definitions
The rank of a linear map is the dimension of its image. It measures how many independent directions in the codomain are reached by the map.
The nullity is the dimension of the kernel. It measures how many independent directions in the domain collapse to zero. Together, rank and nullity divide the domain into the part preserved in the image and the part annihilated by the map.
2 Intuitive meaning
The rank–nullity theorem can be understood as a conservation law for dimension. A linear map may compress some directions, but it cannot change the total number of independent degrees of freedom in its domain without accounting for where those degrees go.
2.1 Geometric interpretation
Geometrically, a linear map can stretch, flatten, or collapse directions. Some vectors may be sent to the same output, and some may disappear entirely. The kernel records the directions that collapse to zero, while the image records the directions that survive as distinct outputs.
For example, a linear map from three-dimensional space to a plane might project one dimension away. In that case, the rank is 2 and the nullity is 1, so the theorem gives \(3=2+1\).
2.2 Relationship between image and kernel
The kernel and image describe complementary aspects of a linear transformation. The kernel identifies redundancies in the input, while the image measures the effective output space.
Although they live in different spaces, they are connected through the action of the map. The theorem says that every dimension in the domain is accounted for by either surviving as part of the image or vanishing in the kernel.
2.3 Dimension counting
The theorem is ultimately a counting argument. Start with a basis for the kernel, then extend it to a basis for the whole domain. The extra basis vectors map to a spanning set of the image, and their number gives the rank.
Thus, the total number of basis vectors in the domain equals the number used by the kernel plus the number needed to describe the image.
3 Proofs
Several standard proofs show the same identity from different viewpoints. Each proof emphasizes a distinct feature of linear maps: bases, quotient spaces, or matrices.
3.1 Proof using a basis of the kernel
Let \(K=\ker(T)\), and choose a basis for \(K\). Because \(V\) is finite-dimensional, this basis can be extended to a basis of all of \(V\). The additional basis vectors, together with those from \(K\), form a complete basis of the domain.
When the added basis vectors are mapped under \(T\), their images span \(\operatorname{im}(T)\). Moreover, these images are linearly independent, so they form a basis of the image. The number of kernel basis vectors is the nullity, and the number of added vectors is the rank. Their sum equals \(\dim(V)\).
3.2 Proof using the first isomorphism theorem
The first isomorphism theorem states that \(V/\ker(T)\) is isomorphic to \(\operatorname{im}(T)\). Taking dimensions gives
\[ \dim(V/\ker(T))=\dim(\operatorname{im}(T)). \]
Since the dimension of a quotient space satisfies
\[ \dim(V/\ker(T))=\dim(V)-\dim(\ker(T)), \]
one obtains
\[ \dim(V)-\operatorname{nullity}(T)=\operatorname{rank}(T), \]
which rearranges to the rank–nullity formula.
3.3 Proof for matrices
For an \(m\times n\) matrix \(A\), row reduction reveals pivot columns and free variables. The number of pivot columns equals the rank, because pivot columns correspond to independent columns in the column space.
The number of free variables equals the nullity, because each free variable contributes one degree of freedom to the solution set of \(Ax=0\). Since every variable is either pivot-related or free, the total number of variables \(n\) equals rank plus nullity.
4 Consequences and corollaries
The theorem has immediate consequences for the behavior of linear maps. It gives simple criteria for injectivity, surjectivity, and invertibility, and it describes the size of solution spaces.
4.1 Injective linear maps
A linear map is injective exactly when its kernel contains only the zero vector. In finite dimensions, this means the nullity is zero.
By rank–nullity, if the nullity is zero then the rank equals the dimension of the domain. Thus an injective map must have full rank relative to its domain.
4.2 Surjective linear maps
A linear map is surjective when its image equals the entire codomain. If the codomain has the same dimension as the domain, surjectivity forces the rank to match the domain dimension.
In finite-dimensional settings, a surjective map from a space to another of the same dimension must also be injective, since there is no room for a nontrivial kernel.
4.3 Criteria for invertibility
A linear map between finite-dimensional vector spaces of equal dimension is invertible if and only if it is injective, if and only if it is surjective. Rank–nullity ties these conditions together.
For a square matrix, invertibility is equivalent to having full rank, which is also equivalent to having nullity zero.
4.4 Dimension of solution spaces
The theorem is often used to determine the number of degrees of freedom in a linear system. For a homogeneous system \(Ax=0\), the dimension of the solution space is the nullity of \(A\).
If the rank is known, the theorem gives the number of independent solutions immediately. This makes it a standard tool for describing families of solutions to linear equations.
5 Applications
Rank–nullity appears throughout linear algebra and related fields. It is especially useful whenever one needs to compare the number of unknowns with the number of independent constraints.
5.1 Solving linear systems
For a system of linear equations, row reduction identifies pivot variables and free variables. Rank counts the independent constraints, while nullity counts the adjustable parameters in the homogeneous case.
This information helps determine whether a system has no solution, a unique solution, or infinitely many solutions. It also clarifies the structure of the full solution set when the system is consistent.
5.2 Matrix rank computations
The theorem supports practical rank calculations by linking rank to nullity. Once the number of free variables in \(Ax=0\) is known, the rank follows from the number of columns.
Conversely, if the rank is obtained from row-echelon form, the nullity is found by subtracting from the total number of variables. This is one of the most efficient ways to analyze a matrix.
5.3 Linear differential equations
In linear differential equations, related dimension-counting ideas appear in the study of solution spaces and linear operators on function spaces. Although the classical theorem is finite-dimensional, its spirit helps explain why the dimension of a space of solutions is tied to the number of independent conditions imposed.
This viewpoint is common in the analysis of boundary value problems and differential operators in more advanced settings.
5.4 Vector space decompositions
The theorem often underlies direct sum decompositions of vector spaces. A basis adapted to the kernel can be extended so that the domain splits into parts that map trivially and parts that contribute to the image.
Such decompositions are useful in theory and computation, since they separate the “collapsed” directions from the directions that determine the output.
6 Related concepts
The rank–nullity theorem is built from a small collection of central ideas in linear algebra. Understanding these ideas makes the theorem more transparent and easier to apply.
6.1 Kernel
The kernel of a linear map is the set of vectors sent to zero. It is always a subspace of the domain.
Its dimension is the nullity. A larger kernel means more input vectors are identified by the map, and therefore less information survives.
6.2 Image
The image is the set of all outputs produced by the linear map. It is a subspace of the codomain.
Its dimension is the rank. The image measures how much of the codomain is actually attained.
6.3 Rank
Rank is the dimension of the image of a linear map or, for a matrix, the dimension of its column space. It records the number of independent output directions.
In many computations, rank is found by counting pivot positions after row reduction.
6.4 Nullity
Nullity is the dimension of the kernel. It measures the number of independent directions that are mapped to zero.
For a matrix, nullity corresponds to the number of free variables in the homogeneous system.
6.5 Dimension theorem in module theory
In broader algebraic settings, similar dimension-counting statements appear for modules and quotient structures. The exact form depends on the algebraic context and on whether a meaningful notion of dimension is available.
The rank–nullity theorem is one of the clearest finite-dimensional examples of this general principle.
7 Generalizations
The finite-dimensional theorem has analogues and extensions in several advanced areas of mathematics. These generalizations preserve the idea that a map’s failure to be invertible can often be measured by complementary quantities.
7.1 Infinite-dimensional settings
In infinite-dimensional vector spaces, the simple identity \(\dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T)\) may no longer be useful in the same way, because dimensions may be infinite cardinals. Special care is needed when comparing sizes of subspaces.
Nevertheless, many operator-theoretic results still reflect a rank–nullity philosophy, especially when kernels and images are finite-dimensional or suitably controlled.
7.2 Fredholm operators
Fredholm operators provide an important extension in functional analysis. For these operators, both the kernel and cokernel are finite-dimensional, and an index can be defined from their difference.
This index theory generalizes the role of rank and nullity by capturing how far an operator is from being invertible in infinite-dimensional contexts.
7.3 Exact sequences
Exact sequences organize linear maps so that the image of one map equals the kernel of the next. In such sequences, dimension relations generalize rank–nullity by balancing the dimensions of successive spaces.
These relations are central in many branches of algebra, where they help track how information passes through a chain of maps.
7.4 Homological algebra perspective
Homological algebra studies structures built from kernels, images, and quotients. From this viewpoint, rank–nullity is an early example of a broader pattern in which algebraic invariants are computed by comparing subspaces and their quotients.
The theorem can therefore be seen as a special case of more general exactness and dimension arguments that appear throughout modern algebra.
</INTERNAL_LINK_CANDIDATES> Kernel (the subspace of vectors mapped to zero) Image (the subspace of outputs attained by the map) Rank (the dimension of the image) Nullity (the dimension of the kernel) Linear transformation (a map preserving addition and scalar multiplication) Vector space (the domain or codomain on which the map acts) Finite-dimensional vector space (a vector space with a finite basis) Matrix (an array representing a linear transformation) Column space (the span of the columns of a matrix) Homogeneous system (a linear system with zero right-hand side) Basis (a set of vectors that spans a space and is linearly independent) Dimension (the number of vectors in a basis) First isomorphism theorem (the result identifying a quotient with the image) Quotient space (the space of cosets modulo a subspace) Injective map (a linear map with trivial kernel) Surjective map (a linear map whose image equals the codomain) Invertible map (a linear map with a two-sided inverse) Row reduction (the process used to find pivots and free variables) Fredholm operator (an operator with finite-dimensional kernel and cokernel) Exact sequence (a chain of maps where images match kernels)