Rank–nullity theorem
Domain dimension equals rank plus nullity.
The rank–nullity theorem is a fundamental result in linear algebra concerning the relationship between the domain, image, and kernel of a linear transformation. It asserts that for a linear transformation between two vector spaces where the domain is finite-dimensional, the dimension of the domain equals the sum of the rank (the dimension of the image) and the nullity (the dimension of the kernel). For matrices, this translates to the number of columns of a matrix being equal to the sum of its rank and nullity. A direct consequence is that for linear transformations between vector spaces of the same finite dimension, injectivity or surjectivity alone implies bijectivity. The theorem can be refined via the splitting lemma to describe an isomorphism of spaces, not just dimensions; because the transformation induces an isomorphism from the domain modulo the kernel to the image, extending a basis of the kernel to a basis of the domain yields the dimension relationship. Two proofs are commonly given: one works generally with linear maps, using the Steinitz exchange lemma to extend a basis of the kernel to a basis of the domain and then showing the images of the extension vectors form a basis for the image. The second proof examines the homogeneous system represented by a matrix, explicitly constructing a set of linearly independent solutions that span the null space, thereby demonstrating that the nullity equals the number of columns minus the rank. While the theorem requires the domain to be finite-dimensional, the codomain need not be; the image itself is finite-dimensional, so the map can be represented by a matrix from the domain to the image. A third fundamental subspace, the cokernel (the quotient of the codomain by the image), is sometimes considered alongside the image and kernel; its dimension, combined with the rank–nullity theorem, forms part of what is called the fundamental theorem of linear algebra. The theorem is a specific instance of the first isomorphism theorem for vector spaces and generalizes to the splitting lemma; it can also be phrased as stating that every short exact sequence of vector spaces splits. In the finite-dimensional case, this formulation extends to exact sequences of any length, where alternating sums of dimensions vanish.
- field
- Linear algebra
- known_for
- Relating the dimensions of the domain, kernel, and image of a linear transformation
Lore & Background
The theorem applies to linear transformations T: V → W where V is finite-dimensional. It states that rank(T) + nullity(T) = dim V. For an m × n matrix M representing a linear map from F^n to F^m, the theorem becomes rank(M) + nullity(M) = n. The theorem can be refined via the splitting lemma to an isomorphism of spaces: Im(T) ⊕ Ker(T) ≅ V. Taking dimensions yields the rank–nullity theorem. Two proofs are given: one using linear maps and basis extension via the Steinitz exchange lemma, and another using the homogeneous system Ax = 0 to show n − r linearly independent solutions span the null space. The theorem requires the domain to be finite-dimensional but places no such assumption on the codomain. Thus, linear maps not given by matrices can still satisfy the theorem, though the first proof is not more general than the second because the image is finite-dimensional.
Reader's Guide
The rank–nullity theorem is a cornerstone of linear algebra, providing a direct relationship between the dimensions of a linear transformation's domain, kernel, and image. It implies that for linear transformations between vector spaces of equal finite dimension, injectivity or surjectivity alone guarantees bijectivity. The theorem is essential for understanding the structure of linear systems, as it links the number of columns of a matrix to its rank and nullity. Its proof via basis extension and the splitting lemma highlights deeper algebraic structure, showing that the image and kernel together form a direct sum isomorphic to the domain. The theorem's applicability extends beyond matrices to any linear map with a finite-dimensional domain, making it a versatile tool in both theoretical and applied contexts.
Did You Know?
- The theorem asserts that the number of columns of a matrix M is the sum of the rank of M and the nullity of M.
- For linear transformations, the dimension of the domain equals the sum of the rank and nullity.
- The theorem can be refined via the splitting lemma to an isomorphism Im(T) ⊕ Ker(T) ≅ V.
- The theorem requires the domain to be finite-dimensional but does not require the codomain to be finite-dimensional.
Frequently Asked Questions
Who is the Rank–nullity theorem?
It is a foundational identity in linear algebra that ties together three dimensions of any linear map: the size of the domain, the size of the kernel (nullity), and the size of the image (rank). In fan-encyclopedia terms, it is the character every linear transformation must carry in its pocket.
What are the Rank–nullity theorem's powers or role?
Its single superpower is the equation dim(domain) = rank + nullity, which says every basis vector in the domain either lands in a new direction of the image or gets annihilated into the kernel. Translated to matrices, the column count always splits into independent column directions plus the dimension of the Ax = 0 solution space.
How does the Rank–nullity theorem's story end?
It always resolves to the same tidy line: the domain's dimension is exactly the sum of the rank and the nullity, with no loose ends. There is no twist or sequel—just a clean accounting of where every vector in the domain goes.
Why is the Rank–nullity theorem important?
It lets you determine whether a map is injective, surjective, or an isomorphism by checking a single dimension count rather than tracking every vector individually. That one-line bookkeeping is why it becomes a go-to tool from first-year linear algebra all the way through functional analysis.
What field does the Rank–nullity theorem call home?
It belongs squarely to linear algebra, specifically the theory of finite-dimensional vector spaces and the linear maps between them. It is typically introduced in a first or second linear-algebra course and then reused constantly in abstract algebra, numerical methods, and applied mathematics.
More in Algebra 1-20
Spotted an error? Know more?
This is a living reference — every entry is fact-audited, and reader corrections feed straight into our audit queue. Suggest an edit · See this site's audit record