Rank–nullity theorem
Domain dimension equals rank plus nullity.
The rank–nullity theorem is a fundamental result in linear algebra. It asserts that for a linear transformation between vector spaces with a finite-dimensional domain, the dimension of the domain equals the sum of the rank (dimension of the image) and the nullity (dimension of the kernel). For matrices, the number of columns equals the sum of the rank and nullity of the matrix.
- 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