# Kernel (linear algebra)

In linear algebra, the kernel of a linear map is the set of all vectors in its domain that the map sends to the zero vector of the codomain. Given a linear map \( T:V \to W \), its kernel is defined as \( \ker T = \{v \in V : T(v) = 0\} \), and this set always forms a linear subspace of the domain \( V \).<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup><sup> • </sup><sup>[2](https://maths.free/linear-algebra/kernel-linear-algebra/)</sup> The kernel is also called the null space or nullspace of the map.<sup>[2](https://maths.free/linear-algebra/kernel-linear-algebra/)</sup><sup> • </sup><sup>[3](https://mathworld.wolfram.com/LinearTransformationKernel.html)</sup> Together with the image (the set of all values the map actually reaches), the kernel measures what a linear map collapses and what it preserves, and the two are tied together by the rank–nullity theorem.<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup>

| Key fact | Detail |
|---|---|
| Definition | For a linear map \( T:V \to W \), \( \ker T = \{v \in V : T(v) = 0\} \); it is a subspace of the domain.<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup> |
| Matrix form | For \( T(x) = Ax \), the kernel is the null space of \( A \) and the range is the column space of \( A \).<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup> |
| Nullity | The dimension of the kernel, \( \operatorname{nullity}(T) = \dim \ker T \).<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup><sup> • </sup><sup>[4](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)</sup> |
| Rank–nullity theorem | If \( V \) is finite dimensional, \( \dim V = \operatorname{rank}(T) + \operatorname{nullity}(T) \); for an \( m \times n \) matrix \( A \), \( \dim(\ker A) + \dim(\operatorname{im} A) = n \).<sup>[5](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web-tacoma/sec-rank-nullity.html)</sup><sup> • </sup><sup>[4](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)</sup> |
| Injectivity test | A linear transformation is injective if and only if its kernel contains only the zero vector.<sup>[6](https://proofwiki.org/wiki/First_Isomorphism_Theorem/Vector_Spaces)</sup> |
| First isomorphism theorem | \( V/\ker T \cong \operatorname{im} T \) via \( v + \ker T \mapsto T(v) \).<sup>[7](https://math.dartmouth.edu/~trs/PreTeXtProjects/linear-algebra-refresher-CLI/output/web/section-factoring-through-a-quotient.html)</sup> |
| Computation | SciPy's `scipy.linalg.null_space` constructs an orthonormal basis for the null space of a matrix using the singular value decomposition.<sup>[8](https://docs.scipy.org/doc/scipy-1.16.1/reference/generated/scipy.linalg.null_space.html)</sup> |

## Definition and basic properties

The kernel collects exactly those inputs on which the map loses all information: every vector \( v \) with \( T(v) = 0 \) becomes indistinguishable from the zero vector, and vectors that differ by an element of the kernel are mapped to the same output.<sup>[2](https://maths.free/linear-algebra/kernel-linear-algebra/)</sup><sup> • </sup><sup>[9](https://watanabe.you/wiki/Kernel%20%28Linear%20Algebra%29)</sup> Because \( T \) is linear, this set is closed under addition and scalar multiplication, so it is a subspace of the domain; the image is correspondingly a subspace of the codomain.<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup><sup> • </sup><sup>[5](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web-tacoma/sec-rank-nullity.html)</sup>

When the map is given by a matrix, \( T(x) = Ax \), the kernel is the null space of \( A \), the solution set of the homogeneous system \( Ax = 0 \), and the image is the column space of \( A \).<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup> Each nonzero vector in the null space corresponds to an eigenvector of the transformation matrix with eigenvalue 0.<sup>[10](https://mathworld.wolfram.com/NullSpace.html)</sup>

## Rank and nullity

The dimension of the kernel is the nullity, and the dimension of the image is the rank.<sup>[1](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)</sup><sup> • </sup><sup>[4](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)</sup> The rank–nullity theorem, also called the dimension theorem, states that for a linear map \( T:V \to W \) with \( V \) finite dimensional,

\[ \dim V = \operatorname{rank}(T) + \operatorname{nullity}(T) = \dim \operatorname{im}(T) + \dim \ker(T). \]<sup>[5](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web-tacoma/sec-rank-nullity.html)</sup>

For an \( m \times n \) matrix \( A \) this reads \( \operatorname{nullity}(A) + \operatorname{rank}(A) = n \), where \( n \) is the number of columns.<sup>[4](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)</sup> The theorem expresses a trade-off: the larger the kernel, the smaller the image, and vice versa.<sup>[4](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)</sup> A standard proof extends a basis \( \mathbf{k}_1, \ldots, \mathbf{k}_m \) of \( \ker f \) to a basis \( \mathbf{k}_1, \ldots, \mathbf{k}_m, \mathbf{u}_1, \ldots, \mathbf{u}_n \) of the domain and shows that the images \( f(\mathbf{u}_1), \ldots, f(\mathbf{u}_n) \) form a basis of the image.<sup>[11](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)</sup>

The kernel also gives a direct test for injectivity: a linear transformation is injective precisely when its kernel contains only the zero vector, since two inputs map to the same output exactly when their difference lies in the kernel.<sup>[6](https://proofwiki.org/wiki/First_Isomorphism_Theorem/Vector_Spaces)</sup>

## Quotients and the first isomorphism theorem

The kernel determines how a map factors through a quotient. The first isomorphism theorem for vector spaces says that for a linear map \( T:V \to W \),

\[ V/\ker T \cong \operatorname{im} T, \]

with the isomorphism given by \( T_*(v + \ker T) = T(v) \); if \( T \) is surjective, then \( V/\ker T \cong W \).<sup>[7](https://math.dartmouth.edu/~trs/PreTeXtProjects/linear-algebra-refresher-CLI/output/web/section-factoring-through-a-quotient.html)</sup> In the quotient \( V/\ker T \), the coset \( v + \ker T \) collects together all elements of \( V \) that map to \( T(v) \), so the theorem identifies the part of the domain the map collapses with the part of the codomain it reaches.<sup>[7](https://math.dartmouth.edu/~trs/PreTeXtProjects/linear-algebra-refresher-CLI/output/web/section-factoring-through-a-quotient.html)</sup><sup> • </sup><sup>[9](https://watanabe.you/wiki/Kernel%20%28Linear%20Algebra%29)</sup>

A concrete example from multivariable calculus: for \( a, b, c \) real numbers not all zero, the map \( T:\mathbb{R}^3 \to \mathbb{R} \) given by \( T(x,y,z) = ax + by + cz \) has as its kernel the plane \( ax + by + cz = 0 \) through the origin. Since \( T \) is surjective, the first isomorphism theorem gives \( \mathbb{R}^3/P \cong \mathbb{R} \), where \( P = \ker T \).<sup>[7](https://math.dartmouth.edu/~trs/PreTeXtProjects/linear-algebra-refresher-CLI/output/web/section-factoring-through-a-quotient.html)</sup>

## Kernels beyond matrices

The same definition applies to linear maps between infinite-dimensional spaces. For a linear differential operator \( L[x, \mathrm{D}] \), the set of all solutions of the homogeneous equation \( L[x, \mathrm{D}]\,y = 0 \) is called the kernel (or nullspace) of the operator; for the first-order operator \( \mathrm{D} - x^2 \), the kernel consists of all multiples of \( e^{x^3/3} \).<sup>[12](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch4/linoperator.html)</sup> A practical consequence is that solutions of equations of the form \( Lu = f \) are not unique when the null space contains more than the zero element, since adding any nontrivial null function to a particular solution yields another solution.<sup>[13](https://www.ideals.illinois.edu/items/11380)</sup>

The kernel concept extends across algebra with the same shape: the kernel of a group homomorphism is a normal subgroup, the kernel of a ring homomorphism is a two-sided ideal, and the kernel of a linear map is a vector subspace.<sup>[14](https://math.dartmouth.edu/~trs/PreTeXtProjects/css-playground/section-4.html)</sup> For a group homomorphism the kernel is the inverse image of the identity element, and for a ring homomorphism the inverse image of zero.<sup>[9](https://watanabe.you/wiki/Kernel%20%28Linear%20Algebra%29)</sup>

## Computation

Numerical software computes kernels through matrix factorizations. SciPy's `null_space` function constructs an orthonormal basis for the null space of a matrix using the singular value decomposition, treating singular values smaller than a threshold \( \texttt{rcond} \cdot \max(s) \) as zero.<sup>[8](https://docs.scipy.org/doc/scipy-1.16.1/reference/generated/scipy.linalg.null_space.html)</sup> The Wolfram Language command `NullSpace[{v1, v2, ...}]` returns a list of vectors forming a basis for the nullspace over the rationals or over whatever base field contains the input vectors.<sup>[10](https://mathworld.wolfram.com/NullSpace.html)</sup>

## History and terminology

The word kernel appears in several distinct mathematical senses. An unrelated use, the integral kernel, occurs in Hilbert's 1904 *Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen* in German and in Bochner's 1909 introduction to integral equations in English; the algebraic sense seems to have arisen in group theory before linear algebra or ring theory, and [Lev Pontryagin](https://www.edgepedia.org/lev-pontryagin) used the term "Kern des Homomorphismus" in his 1931 paper in *Mathematische Annalen*.<sup>[15](https://hsm.stackexchange.com/questions/2149/what-came-first-the-kernel-from-vector-spaces-or-from-group-theory?noredirect=1)</sup> In subsequent systematic treatments of linear and abstract algebra, [Hermann Weyl](https://www.edgepedia.org/hermann-weyl) and [Bartel Leendert van der Waerden](https://www.edgepedia.org/bartel-leendert-van-der-waerden) integrated kernels into their frameworks, and the notion thereby ceased to depend on any specific matrix representation.<sup>[9](https://watanabe.you/wiki/Kernel%20%28Linear%20Algebra%29)</sup>

## References

1. [Kernel and Range Cheat Sheet, Math101.ca](https://learn.math101.ca/cheatsheets/linear-algebra-kernel-and-range/)
2. [Kernel (linear algebra), maths.free](https://maths.free/linear-algebra/kernel-linear-algebra/)
3. [Linear Transformation Kernel, Wolfram MathWorld](https://mathworld.wolfram.com/LinearTransformationKernel.html)
4. [Applied Linear Algebra, Chapter 3 (Rank–Nullity Theorem), NCU](https://staff.csie.ncu.edu.tw/chia/Course/LinearAlgebra/download/chap3-main.pdf)
5. [Rank and Nullity, Linear Algebra Companion, Dartmouth](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web-tacoma/sec-rank-nullity.html)
6. [First Isomorphism Theorem/Vector Spaces, ProofWiki](https://proofwiki.org/wiki/First_Isomorphism_Theorem/Vector_Spaces)
7. [Linear Maps out of Quotients, Linear Algebra Refresher, Dartmouth](https://math.dartmouth.edu/~trs/PreTeXtProjects/linear-algebra-refresher-CLI/output/web/section-factoring-through-a-quotient.html)
8. [scipy.linalg.null_space, SciPy v1.16.1 Manual](https://docs.scipy.org/doc/scipy-1.16.1/reference/generated/scipy.linalg.null_space.html)
9. [Kernel (Linear Algebra), watanabe.you wiki](https://watanabe.you/wiki/Kernel%20%28Linear%20Algebra%29)
10. [Null Space, Wolfram MathWorld](https://mathworld.wolfram.com/NullSpace.html)
11. [Linear Maps, MATH0007 Algebra for Joint Honours Students, UCL](https://www.homepages.ucl.ac.uk/~ucahmto/0007/_book/3-3-linear-maps.html)
12. [Mathematica Tutorial, Part 1.4: Linear Operators, Brown University](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch4/linoperator.html)
13. [Null-Space Methods for Numerical Solutions of Differential Equations, IDEALS, University of Illinois](https://www.ideals.illinois.edu/items/11380)
14. [A Fundamental Isomorphism Theorem for Groups, Rings, Vector Spaces, Dartmouth](https://math.dartmouth.edu/~trs/PreTeXtProjects/css-playground/section-4.html)
15. [What came first? The kernel from vector spaces or from group theory?, History of Science and Mathematics Stack Exchange](https://hsm.stackexchange.com/questions/2149/what-came-first-the-kernel-from-vector-spaces-or-from-group-theory?noredirect=1)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Vector spaces and linear maps*

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