Kernel (linear algebra)
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In linear algebra and functional analysis, the kernel (also null space or nullspace) of a linear map L : V → W between two vector spaces or two modules V and W is the set of all elements v of V for which L(v) = 0. That is
where 0 denotes the zero vector in W. The kernel of L is a linear subspace of the domain V.^{1} For a linear map given as a matrix A, the kernel is simply the set of solutions to the equation , where x and 0 are understood to be column vectors. The dimension of the null space of A is called the nullity of A.
Contents
Definition
The kernel of a m × n matrix A with coefficients in a field K (typically the field of the real numbers or of the complex numbers) is the set
 ^{2}
where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations:
From this viewpoint, the null space of A is the same as the solution set to the homogeneous system.
The same definition fits for matrices over a ring K, replacing the "vector nspace" with "right free module" and a "linear subspace" with "submodule", but there is no concept of nullity.
Computation
In this section, we show on a very simple example how the null space of a matrix may be computed. However the method which is sketched here is not practical for effective computations. A more efficient method is presented below.
Consider the matrix
The null space of this matrix consists of all vectors (x, y, z) ∈ R^{3} for which
This can be written as a homogeneous system of linear equations involving x, y, and z:
This can be written in matrix form as:
Using Gauss–Jordan elimination, this reduces to:
Rewriting yields:
Now we can write the null space (solution to Ax = 0) in terms of c, where c is scalar:
Since c is a free variable this can be simplified to
The null space of A is precisely the set of solutions to these equations (in this case, a line through the origin in R^{3}).
Examples
 If L: R^{m} → R^{n}, then the kernel of L is the solution set to a homogeneous system of linear equations. For example, if L is the operator:

 then the kernel of L is the set of solutions to the equations
 Let C[0,1] denote the vector space of all continuous realvalued functions on the interval [0,1], and define L: C[0,1] → R by the rule

 Then the kernel of L consists of all functions f ∈ C[0,1] for which f(0.3) = 0.
 Let C^{∞}(R) be the vector space of all infinitely differentiable functions R → R, and let D: C^{∞}(R) → C^{∞}(R) be the differentiation operator:

 Then the kernel of D consists of all functions in C^{∞}(R) whose derivatives are zero, i.e. the set of all constant functions.
 Let R^{∞} be the direct product of infinitely many copies of R, and let s: R^{∞} → R^{∞} be the shift operator

 Then the kernel of s is the onedimensional subspace consisting of all vectors (x_{1}, 0, 0, ...).
 If V is an inner product space and W is a subspace, the kernel of the orthogonal projection V → W is the orthogonal complement to W in V.
Subspace properties
The null space of an m × n real matrix is a linear subspace of R^{n}. That is, the set Null(A) has the following three properties:
 Null(A) always contains the zero vector, since A0 = 0.
 If x ∈ Null(A) and y ∈ Null(A), then x + y ∈ Null(A). This follows from the distributivity of matrix multiplication over addition.
 If x ∈ Null(A) and c is a scalar, then cx ∈ Null(A), since A(cx) = c(Ax) = c0 = 0.
If L: V → W, then two elements of V have the same image in W if and only if their difference lies in the kernel of L:
It follows that the image of L is isomorphic to the quotient of V by the kernel:
This implies the ranknullity theorem:
When V is an inner product space, the quotient V / ker(L) can be identified with the orthogonal complement in V of ker(L). This is the generalization to linear operators of the row space of a matrix.
Basis
A basis of the null space of a matrix may be computed by Gaussian elimination.
For this purpose, given an m × n matrix A, we construct first the row augmented matrix where I is the n × n identity matrix.
Computing its column echelon form by Gaussian elimination (or any other available method), we get a matrix A basis of the null space of A consists in the nonzero columns of C such that the corresponding column of B is a zero column.
In fact, the computation may be stopped as soon as the upper matrix is in column echelon form: the remainder of the computation consists in changing the basis of the vector space generated by the columns whose upper part is zero.
For example, suppose that
Then
Putting the upper part in column echelon form by column operations on the whole matrix gives
The last three columns of B are zero columns. Therefore, the three last vectors of C,
are a basis of the null space of A.
Relation to the row space
Let A be an m by n matrix (i.e., A has m rows and n columns). The product of A and the ndimensional vector x can be written in terms of the dot product of vectors as follows:
Here a_{1}, ... , a_{m} denote the rows of the matrix A. It follows that x is in the null space of A if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (because if the dot product of two vectors is equal to zero they are by definition orthogonal).
The row space of a matrix A is the span of the row vectors of A. By the above reasoning, the null space of A is the orthogonal complement to the row space. That is, a vector x lies in the null space of A if and only if it is perpendicular to every vector in the row space of A.
The dimension of the row space of A is called the rank of A, and the dimension of the null space of A is called the nullity of A. These quantities are related by the equation
The equation above is known as the rank–nullity theorem.
Nonhomogeneous equations
The null space also plays a role in the solution to a nonhomogeneous system of linear equations:
If u and v are two possible solutions to the above equation, then
Thus, the difference of any two solutions to the equation Ax = b lies in the null space of A.
It follows that any solution to the equation Ax = b can be expressed as the sum of a fixed solution v and an arbitrary element of the null space. That is, the solution set to the equation Ax = b is
where v is any fixed vector satisfying Av = b. Geometrically, this says that the solution set to Ax = b is the translation of the null space of A by the vector v. See also Fredholm alternative and flat (geometry).
Left null space
The left null space of a matrix A consists of all vectors x such that x^{T}A = 0^{T}, where T denotes the transpose of a column vector. The left null space of A is the same as the null space of A^{T}. The left null space of A is the orthogonal complement to the column space of A, and is the cokernel of the associated linear transformation. The null space, the row space, the column space, and the left null space of A are the four fundamental subspaces associated to the matrix A.
Kernels in functional analysis
If V and W are topological vector spaces (and W is finitedimensional) then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V.
Numerical computation
The problem of computing the null space on a computer depends on the nature of the coefficients.
Exact coefficients
If the coefficients of the matrix are exactly given integers numbers, the column echelon form of the matrix may be computed by Bareiss algorithm more efficiently than with Gaussian elimination. It is even more efficient to use modular arithmetic, which reduces the problem to a similar one over a finite field.
For coefficients in a finite field Gaussian elimination works well, but for the large matrices that occur in cryptography and Gröbner basis computation, better algorithms are known, which have roughly the same computational complexity, but are faster and behave better with modern computer hardware.
Floating point computation
For matrices whose entries are floatingpoint numbers, the problem of computing the null space makes sense only for matrices such that the number of rows is equal to their rank: because of the rounding errors, a floatingpoint matrix has almost always a full rank, even when it is an approximation of a matrix of a much smaller rank. Even for a fullrank matrix, it is possible to compute its null space only if it is well conditioned, i.e. it has a low condition number.
Even for a well conditioned full rank matrix, Gaussian elimination does not behave correctly: it introduces rounding errors that are too large for getting a significant result. As the computation of the null space of a matrix is a special instance of solving a homogeneous system of linear equations, the null space may be computed by any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the Lapack library.
See also
 System of linear equations
 Row and column spaces
 Row reduction
 Four fundamental subspaces
 Vector space
 Linear subspace
 Linear operator
 Function space
 Fredholm alternative
Notes
 ^ Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in Lay 2005, Meyer 2001, and Strang 2005.
 ^ This equation uses setbuilder notation.
References
 Axler, Sheldon Jay (1997), Linear Algebra Done Right (2nd ed.), SpringerVerlag, ISBN 0387982590
 Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 9780321287137
 Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 9780898714548
 Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, ISBN 0534998453
 Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
 Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall
 Serge Lang (1987). Linear Algebra. Springer. p. 59. ISBN 9780387964126.
 Lloyd N. Trefethen and David Bau, III, Numerical Linear Algebra, SIAM 1997, ISBN 9780898713619 online version
External links
Wikibooks has a book on the topic of: Linear Algebra/Null Spaces 
 Hazewinkel, Michiel, ed. (2001), "Kernel of a matrix", Encyclopedia of Mathematics, Springer, ISBN 9781556080104
 Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces at Google Video, from MIT OpenCourseWare
 Khan Academy, Introduction to the Null Space of a Matrix
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