Editing Kernel (linear algebra)

{{Short description|Vectors mapped to 0 by a linear map}}
{{other uses|Kernel (disambiguation)}}
[[File:Projection-on-diagonal.gif|thumb|300x300px|An example for a kernel- the linear operator <math> L : (x,y) \longrightarrow (x, x)</math> transforms all points on the <math> (x=0, y)</math> line to the zero point <math> (0,0)</math>, thus they form the kernel for the linear operator]]
In [[mathematics]], the '''kernel''' of a [[linear map]], also known as the '''null space''' or '''nullspace''', is the part of the [[Domain of a function|domain]] which is mapped to the [[Zero element#Additive identities|zero vector]] of the [[Codomain|co-domain]]; the kernel is always a [[linear subspace]] of the domain.<ref>{{Cite web|url=http://mathworld.wolfram.com/Kernel.html|title=Kernel|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-09}}</ref> That is, given a linear map {{math|''L'' : ''V'' → ''W''}} between two [[vector space]]s {{mvar|V}} and {{mvar|W}}, the kernel of {{mvar|L}} is the vector space of all elements {{math|'''v'''}} of {{mvar|V}} such that {{math|1=''L''('''v''') = '''0'''}}, where {{math|'''0'''}} denotes the [[zero vector]] in {{mvar|W}},<ref name=":0">{{Cite web|url=https://brilliant.org/wiki/kernel/|title=Kernel (Nullspace) {{!}} Brilliant Math & Science Wiki|website=brilliant.org|language=en-us|access-date=2019-12-09}}</ref> or more symbolically:
<math display="block">\ker(L) = \left\{ \mathbf{v} \in V \mid L(\mathbf{v})=\mathbf{0} \right\} = L^{-1}(\mathbf{0}).</math>

==Properties==
[[File:Kernel and image of linear map.svg|thumb|300px|Kernel and image of a linear map {{mvar|L}} from {{mvar|V}} to {{mvar|W}}]]
The kernel of {{mvar|L}} is a [[linear subspace]] of the domain {{mvar|V}}.<ref name="textbooks">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 {{harvnb|Lay|2005}}, {{harvnb|Meyer|2001}}, and Strang's lectures.</ref><ref name=":0" />
In the linear map <math>L : V \to W,</math> two elements of {{mvar|V}} have the same [[Image (mathematics)|image]] in {{mvar|W}} if and only if their difference lies in the kernel of {{mvar|L}}, that is,
<math display=block>L\left(\mathbf{v}_1\right) = L\left(\mathbf{v}_2\right) \quad \text{ if and only if } \quad L\left(\mathbf{v}_1-\mathbf{v}_2\right) = \mathbf{0}.</math>

From this, it follows by the [[Isomorphism_theorems#First_isomorphism_theorem|first isomorphism theorem]] that the image of {{mvar|L}} is [[Vector space isomorphism|isomorphic]] to the [[Quotient space (linear algebra)|quotient]] of {{mvar|V}} by the kernel:
<math display=block>\operatorname{im}(L) \cong V / \ker(L).</math>
{{anchor|nullity}}In the case where {{mvar|V}} is [[finite-dimensional]], this implies the [[rank–nullity theorem]]:
<math display=block>\dim(\ker L) + \dim(\operatorname{im} L) = \dim(V).</math>
where the term {{em|{{visible anchor|rank}}}} refers to the dimension of the image of {{mvar|L}}, <math>\dim(\operatorname{im} L),</math> while ''{{em|{{visible anchor|nullity}}}}'' refers to the dimension of the kernel of {{mvar|L}}, <math>\dim(\ker L).</math><ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|url=http://mathworld.wolfram.com/Rank-NullityTheorem.html|title=Rank-Nullity Theorem|website=mathworld.wolfram.com|language=en|access-date=2019-12-09}}</ref> 
That is,
<math display=block>\operatorname{Rank}(L) = \dim(\operatorname{im} L) \qquad \text{ and } \qquad \operatorname{Nullity}(L) = \dim(\ker L),</math>
so that the rank–nullity theorem can be restated as 
<math display=block>\operatorname{Rank}(L) + \operatorname{Nullity}(L) = \dim \left(\operatorname{domain} L\right).</math>

When {{mvar|V}} is an [[inner product space]], the quotient <math>V / \ker(L)</math> can be identified with the [[orthogonal complement]] in {{mvar|V}} of <math>\ker(L)</math>. This is the generalization to linear operators of the [[row space]], or coimage, of a matrix.

==Generalization to modules==

{{main|Module (mathematics)}}
The notion of kernel also makes sense for [[homomorphism]]s of [[Module (mathematics)|modules]], which are generalizations of vector spaces where the scalars are elements of a [[Ring (mathematics)|ring]], rather than a [[Field (mathematics)|field]]. The domain of the mapping is a module, with the kernel constituting a [[submodule]]. Here, the concepts of rank and nullity do not necessarily apply.

==In functional analysis==

{{main|Topological vector space}}
If {{mvar|V}} and {{mvar|W}} are [[topological vector space]]s such that {{mvar|W}} is finite-dimensional, then a linear operator {{math|''L'': ''V'' → ''W''}} is [[continuous linear operator|continuous]] if and only if the kernel of {{mvar|L}} is a [[closed set|closed]] subspace of {{mvar|V}}.

==Representation as matrix multiplication==
Consider a linear map represented as a {{math|''m'' × ''n''}} matrix {{mvar|A}} with coefficients in a [[field (mathematics)|field]] {{mvar|K}} (typically <math>\mathbb{R}</math> or <math>\mathbb{C}</math>), that is operating on column vectors {{math|'''x'''}} with {{mvar|n}} components over {{mvar|K}}.
The kernel of this linear map is the set of solutions to the equation {{math|1=''A'''''x''' = '''0'''}}, where {{math|'''0'''}} is understood as the [[zero vector]]. The [[dimension (vector space)|dimension]] of the kernel of ''A'' is called the '''nullity''' of ''A''. In [[set-builder notation]],
<math display="block">\operatorname{N}(A) = \operatorname{Null}(A) = \operatorname{ker}(A) = \left\{ \mathbf{x}\in K^n \mid A\mathbf{x} = \mathbf{0} \right\}.</math>
The matrix equation is equivalent to a homogeneous [[system of linear equations]]:
<math display="block">A\mathbf{x}=\mathbf{0} \;\;\Leftrightarrow\;\;
\begin{alignat}{7}
a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& 0      \\
a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& 0      \\
           &&       &&            &&                    &&            &&\vdots\ \;&&&    \\
a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& 0\text{.}      \\
\end{alignat}</math>
Thus the kernel of ''A'' is the same as the solution set to the above homogeneous equations.

===Subspace properties===
The kernel of a {{math|''m'' × ''n''}} matrix {{mvar|A}} over a field {{mvar|K}} is a [[linear subspace]] of {{math|'''K'''<sup>''n''</sup>}}. That is, the kernel of {{mvar|A}}, the set {{math|Null(''A'')}}, has the following three properties:
# {{math|Null(''A'')}} always contains the [[zero vector]], since {{math|1=''A'''''0''' = '''0'''}}.
# If {{math|'''x''' ∈ Null(''A'')}} and {{math|'''y''' ∈ Null(''A'')}}, then {{math|'''x''' + '''y''' ∈ Null(''A'')}}. This follows from the distributivity of matrix multiplication over addition.
# If {{math|'''x''' ∈ Null(''A'')}} and {{mvar|c}} is a [[scalar (mathematics)|scalar]] {{math|''c'' ∈ ''K''}}, then {{math|''c'''''x''' ∈ Null(''A'')}}, since {{math|1=''A''(''c'''''x''') = ''c''(''A'''''x''') = ''c'''''0''' = '''0'''}}.

===The row space of a matrix===
{{main|Rank–nullity theorem}}
The product ''A'''''x''' can be written in terms of the [[dot product]] of vectors as follows:
<math display="block">A\mathbf{x} = \begin{bmatrix} \mathbf{a}_1 \cdot \mathbf{x} \\ \mathbf{a}_2 \cdot \mathbf{x} \\ \vdots \\ \mathbf{a}_m \cdot \mathbf{x} \end{bmatrix}.</math>

Here, {{math|'''a'''<sub>1</sub>, ... , '''a'''<sub>''m''</sub>}} denote the rows of the matrix {{mvar|A}}. It follows that {{math|'''x'''}} is in the kernel of {{mvar|A}}, if and only if {{math|'''x'''}} is [[orthogonality|orthogonal]] (or perpendicular) to each of the row vectors of {{mvar|A}} (since orthogonality is defined as having a dot product of 0).

The [[row space]], or coimage, of a matrix {{mvar|A}} is the [[linear span|span]] of the row vectors of {{mvar|A}}. By the above reasoning, the kernel of {{mvar|A}} is the [[orthogonal complement]] to the row space. That is, a vector {{math|'''x'''}} lies in the kernel of {{mvar|A}}, if and only if it is perpendicular to every vector in the row space of {{mvar|A}}.

The dimension of the row space of {{mvar|A}} is called the [[rank (linear algebra)|rank]] of ''A'', and the dimension of the kernel of {{mvar|A}} is called the '''nullity''' of {{mvar|A}}. These quantities are related by the [[rank–nullity theorem]]<ref name=":1" />
<math display="block">\operatorname{rank}(A) + \operatorname{nullity}(A) = n.</math>

===Left null space===
The '''left null space''', or [[cokernel]], of a matrix {{mvar|A}} consists of all column vectors {{math|'''x'''}} such that {{math|1='''x'''<sup>T</sup>''A'' = '''0'''<sup>T</sup>}}, where T denotes the [[transpose]] of a matrix.  The left null space of {{mvar|A}} is the same as the kernel of {{math|''A''<sup>T</sup>}}. The left null space of {{mvar|A}} is the orthogonal complement to the [[column space]] of {{mvar|A}}, and is dual to the [[cokernel]] of the associated linear transformation. The kernel, the row space, the column space, and the left null space of {{mvar|A}} are the '''four fundamental subspaces''' associated with the matrix {{mvar|A}}.

===Nonhomogeneous systems of linear equations===
The kernel also plays a role in the solution to a nonhomogeneous system of linear equations:
<math display="block">A\mathbf{x} = \mathbf{b}\quad \text{or} \quad \begin{alignat}{7}
a_{11} x_1 &&\; + \;&& a_{12} x_2 &&\; + \;\cdots\; + \;&& a_{1n} x_n &&\; = \;&&& b_1      \\
a_{21} x_1 &&\; + \;&& a_{22} x_2 &&\; + \;\cdots\; + \;&& a_{2n} x_n &&\; = \;&&& b_2      \\
           &&       &&            &&                    &&            &&\vdots\ \;&&&       \\
a_{m1} x_1 &&\; + \;&& a_{m2} x_2 &&\; + \;\cdots\; + \;&& a_{mn} x_n &&\; = \;&&& b_m      \\
\end{alignat}</math>
If {{math|'''u'''}} and {{math|'''v'''}} are two possible solutions to the above equation, then
<math display="block">A(\mathbf{u} - \mathbf{v}) = A\mathbf{u} - A\mathbf{v} = \mathbf{b} - \mathbf{b} = \mathbf{0}</math>
Thus, the difference of any two solutions to the equation {{math|1=''A'''''x''' = '''b'''}} lies in the kernel of {{mvar|A}}.

It follows that any solution to the equation {{math|1=''A'''''x''' = '''b'''}} can be expressed as the sum of a fixed solution {{math|'''v'''}} and an arbitrary element of the kernel. That is, the solution set to the equation {{math|1=''A'''''x''' = '''b'''}} is
<math display="block">\left\{ \mathbf{v}+\mathbf{x} \mid A \mathbf{v}=\mathbf{b} \land \mathbf{x}\in\operatorname{Null}(A) \right\},</math>
Geometrically, this says that the solution set to {{math|1=''A'''''x''' = '''b'''}} is the [[translation (geometry)|translation]] of the kernel of {{mvar|A}} by the vector {{math|'''v'''}}. See also [[Fredholm alternative]] and [[flat (geometry)]].

==Illustration==
The following is a simple illustration of the computation of the kernel of a matrix (see {{slink||Computation by Gaussian elimination}}, below for methods better suited to more complex calculations). The illustration also touches on the row space and its relation to the kernel.

Consider the matrix
<math display="block">A = \begin{bmatrix} 2 & 3 & 5 \\ -4 & 2 & 3 \end{bmatrix}.</math>
The kernel of this matrix consists of all vectors {{math|(''x'', ''y'', ''z'') ∈ [[real coordinate space|'''R'''<sup>3</sup>]]}} for which
<math display="block">\begin{bmatrix} 2 & 3 & 5 \\ -4 & 2 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix},</math>
which can be expressed as a homogeneous [[system of linear equations]] involving {{mvar|x}}, {{mvar|y}}, and {{mvar|z}}:
<math display="block">\begin{align}
 2x + 3y + 5z &= 0, \\
-4x + 2y + 3z &= 0.
\end{align}</math>

The same linear equations can also be written in matrix form as:
<math display="block">
  \left[\begin{array}{ccc|c}
    2 & 3 & 5 & 0 \\
    -4 & 2 & 3 & 0
  \end{array}\right].
</math>

Through [[Gauss–Jordan elimination]], the matrix can be reduced to:
<math display="block">
  \left[\begin{array}{ccc|c}
    1 & 0 & 1/16 & 0 \\
    0 & 1 & 13/8 & 0
  \end{array}\right].
</math>

Rewriting the matrix in equation form yields:
<math display="block">\begin{align}
x &= -\frac{1}{16}z \\
y &= -\frac{13}{8}z.
\end{align}</math>

The elements of the kernel can be further expressed in [[Parametric equations|parametric vector form]], as follows:
<math display="block">\begin{bmatrix} x \\ y \\ z\end{bmatrix} = c \begin{bmatrix} -1/16 \\ -13/8 \\ 1\end{bmatrix}\quad (\text{where }c \in \mathbb{R})</math>

Since {{mvar|c}} is a [[Free variables (system of linear equations)|free variable]] ranging over all real numbers, this can be expressed equally well as:
<math display="block">
\begin{bmatrix} x \\ y \\ z \end{bmatrix}
 = c \begin{bmatrix} -1 \\ -26 \\ 16 \end{bmatrix}.
</math>
The kernel of {{mvar|A}} is precisely the solution set to these equations (in this case, a [[line (geometry)|line]] through the origin in {{math|'''R'''<sup>3</sup>}}). Here, the vector {{math|(−1,−26,16)<sup>T</sup>}} constitutes a [[Basis (linear algebra)|basis]] of the kernel of {{mvar|A}}. The nullity of {{mvar|A}} is therefore 1, as it is spanned by a single vector.

The following dot products are zero:
<math display="block">
 \begin{bmatrix} 2 & 3 & 5 \end{bmatrix}
 \begin{bmatrix} -1 \\ -26 \\ 16 \end{bmatrix}
= 0
\quad\mathrm{and}\quad
 \begin{bmatrix} -4 & 2 & 3 \end{bmatrix}
 \begin{bmatrix} -1 \\ -26 \\ 16 \end{bmatrix}
= 0 ,
</math>
which illustrates that vectors in the kernel of {{mvar|A}} are orthogonal to each of the row vectors of {{mvar|A}}.

These two (linearly independent) row vectors span the row space of {{mvar|A}}—a plane orthogonal to the vector {{math|(−1,−26,16)<sup>T</sup>}}.

With the rank 2 of {{mvar|A}}, the nullity 1 of {{mvar|A}}, and the dimension 3 of {{mvar|A}}, we have an illustration of the rank-nullity theorem.

==Examples==

*If {{math|''L'': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}}, then the kernel of {{math|''L''}} is the solution set to a homogeneous [[system of linear equations]].  As in the above illustration, if {{math|''L''}} is the operator: <math display="block"> L(x_1,  x_2, x_3) = (2 x_1 + 3 x_2 + 5 x_3,\; - 4 x_1 + 2 x_2 + 3 x_3)</math> then the kernel of {{math|''L''}} is the set of solutions to the equations <math display="block"> \begin{alignat}{7}
    2x_1 &\;+\;& 3x_2 &\;+\;& 5x_3 &\;=\;& 0 \\
   -4x_1 &\;+\;& 2x_2 &\;+\;& 3x_3 &\;=\;& 0
\end{alignat}</math>
*Let {{math|''C''[0,1]}} denote the [[vector space]] of all continuous real-valued functions on the interval [0,1], and define {{math|''L'': ''C''[0,1] → '''R'''}} by the rule <math display="block">L(f) = f(0.3).</math> Then the kernel of {{math|''L''}} consists of all functions {{math|1=''f'' ∈ ''C''[0,1]}} for which {{math|1=''f''(0.3) = 0}}.
*Let {{math|''C''<sup>∞</sup>('''R''')}} be the vector space of all infinitely differentiable functions {{math|'''R''' → '''R'''}}, and let {{math|''D'': ''C''<sup>∞</sup>('''R''') → ''C''<sup>∞</sup>('''R''')}} be the [[differential operator|differentiation operator]]: <math display="block">D(f) = \frac{df}{dx}.</math> Then the kernel of {{math|''D''}} consists of all functions in {{math|''C''<sup>∞</sup>('''R''')}} whose derivatives are zero, i.e. the set of all [[constant function]]s.
*Let {{math|'''R'''<sup>∞</sup>}} be the [[direct product]] of infinitely many copies of {{math|'''R'''}}, and let {{math|''s'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>}} be the [[shift operator]] <math display="block"> s(x_1, x_2, x_3, x_4, \ldots) = (x_2, x_3, x_4, \ldots).</math> Then the kernel of {{math|''s''}} is the one-dimensional subspace consisting of all vectors {{math|(''x''<sub>1</sub>, 0, 0, 0, ...)}}.
*If {{mvar|V}} is an [[inner product space]] and {{mvar|W}} is a subspace, the kernel of the [[projection (linear algebra)|orthogonal projection]] {{math|''V'' → ''W''}} is the [[orthogonal complement]] to {{mvar|W}} in {{mvar|V}}.

==Computation by Gaussian elimination==
A [[Basis (linear algebra)|basis]] of the kernel of a matrix may be computed by [[Gaussian elimination]].

For this purpose, given an {{math|''m'' × ''n''}} matrix {{mvar|A}}, we construct first the row [[augmented matrix]] <math> \begin{bmatrix}A \\ \hline I \end{bmatrix},</math> where {{math|''I''}}<!-- necessary to ensure a serif font --> is the {{math|''n'' × ''n''}} [[identity matrix]].

Computing its [[column echelon form]] by Gaussian elimination (or any other suitable method), we get a matrix <math> \begin{bmatrix} B \\\hline C \end{bmatrix}.</math> A basis of the kernel of {{Mvar|A}} consists in the non-zero columns of {{mvar|C}} such that the corresponding column of {{mvar|B}} is a [[zero matrix|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
<math display="block">A = \begin{bmatrix}
1 & 0 & -3 & 0 &  2 & -8 \\
0 & 1 &  5 & 0 & -1 & 4 \\
0 & 0 &  0 & 1 & 7 & -9 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}. </math>
Then
<math display="block"> \begin{bmatrix} A \\ \hline I \end{bmatrix} =
\begin{bmatrix}
1 & 0 & -3 & 0 &  2 & -8 \\
0 & 1 &  5 & 0 & -1 & 4 \\
0 & 0 &  0 & 1 & 7 & -9 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\hline
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{bmatrix}. </math>

Putting the upper part in column echelon form by column operations on the whole matrix gives
<math display="block"> \begin{bmatrix} B \\ \hline C \end{bmatrix} =
\begin{bmatrix}
1 & 0 &  0 & 0 &  0 & 0 \\
0 & 1 &  0 & 0 &  0 & 0 \\
0 & 0 &  1 & 0 &  0 & 0 \\
0 & 0 &  0 & 0 &  0 & 0 \\
\hline
1 & 0 &  0 & 3 & -2 & 8 \\
0 & 1 &  0 & -5 & 1 & -4 \\
0 & 0 &  0 & 1 & 0 & 0 \\
0 & 0 &  1 & 0 & -7 & 9 \\
0 & 0 &  0 & 0 & 1 & 0 \\
0 & 0 &  0 & 0 & 0 & 1
\end{bmatrix}. </math>

The last three columns of {{mvar|B}} are zero columns. Therefore, the three last vectors of {{mvar|C}},
<math display="block">\left[\!\! \begin{array}{r} 3 \\ -5 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right] ,\;
\left[\!\! \begin{array}{r} -2 \\ 1 \\ 0 \\ -7 \\ 1 \\ 0 \end{array} \right],\;
\left[\!\! \begin{array}{r} 8 \\ -4 \\ 0 \\ 9 \\ 0 \\ 1 \end{array} \right] </math>
are a basis of the kernel of {{mvar|A}}.

Proof that the method computes the kernel: Since column operations correspond to post-multiplication by invertible matrices, the fact that <math>\begin{bmatrix} A \\ \hline I \end{bmatrix}</math> reduces to <math>\begin{bmatrix} B \\ \hline C \end{bmatrix}</math> means that there exists an invertible matrix <math>P</math> such that <math> \begin{bmatrix} A \\ \hline I \end{bmatrix} P = \begin{bmatrix} B \\ \hline C \end{bmatrix}, </math> with <math>B</math> in column echelon form. Thus {{nowrap|<math>AP = B</math>,}} {{nowrap|<math>IP = C </math>,}} and {{nowrap|<math> AC = B </math>.}} A column vector <math>\mathbf v</math> belongs to the kernel of <math>A</math> (that is <math>A \mathbf v = \mathbf 0</math>) if and only if <math>B \mathbf w = \mathbf 0,</math> where {{nowrap|<math>\mathbf w = P^{-1} \mathbf v = C^{-1} \mathbf v</math>.}} As <math>B</math> is in column echelon form, {{nowrap|<math>B \mathbf w = \mathbf 0</math>,}} if and only if the nonzero entries of <math>\mathbf w</math> correspond to the zero columns of {{nowrap|<math>B</math>.}} By multiplying by {{nowrap|<math>C</math>,}} one may deduce that this is the case if and only if <math>\mathbf v = C \mathbf w</math> is a linear combination of the corresponding columns of {{nowrap|<math>C</math>.}}

==Numerical computation==
The problem of computing the kernel on a computer depends on the nature of the coefficients.

===Exact coefficients===
If the coefficients of the matrix are exactly given numbers, the [[column echelon form]] of the matrix may be computed with [[Bareiss algorithm]] more efficiently than with Gaussian elimination. It is even more efficient to use [[modular arithmetic]] and [[Chinese remainder theorem]], which reduces the problem to several similar ones over [[finite field]]s (this avoids the overhead induced by the non-linearity of the [[computational complexity]] of integer multiplication).{{Citation needed|date=October 2014}}

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 [[Analysis of algorithms|computational complexity]], but are faster and behave better with modern [[computer hardware]].{{Citation needed|date=October 2014}}

===Floating point computation===
For matrices whose entries are [[floating-point number]]s, the problem of computing the kernel makes sense only for matrices such that the number of rows is equal to their rank: because of the [[rounding error]]s, a floating-point matrix has almost always a [[full rank]], even when it is an approximation of a matrix of a much smaller rank. Even for a full-rank matrix, it is possible to compute its kernel only if it is [[well-conditioned problem|well conditioned]], i.e. it has a low [[condition number]].<ref>{{Cite web |url=https://www.math.ohiou.edu/courses/math3600/lecture11.pdf |title=Archived copy |access-date=2015-04-14 |archive-url=https://web.archive.org/web/20170829031912/http://www.math.ohiou.edu/courses/math3600/lecture11.pdf |archive-date=2017-08-29 |url-status=dead }}</ref>{{Citation needed|date=December 2019}}

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 kernel of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the [[Lapack]] library.{{Citation needed|date=October 2014}}

==See also==
{{Div col|colwidth=20em}}
* [[Kernel (algebra)]]
* [[Zero set]]
* [[System of linear equations]]
* [[Row and column spaces]]
* [[Row reduction]]
* [[Four fundamental subspaces]]
* [[Vector space]]
* [[Linear subspace]]
* [[Linear operator]]
* [[Function space]]
* [[Fredholm alternative]]
{{div col end}}

==Notes and references==
{{reflist}}

==Bibliography==
{{see also|Linear algebra#Further reading}}
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==External links==
{{wikibooks|Linear Algebra/Null Spaces}}
* {{springer|title=Kernel of a matrix|id=p/k110090}}
* [[Khan Academy]], [http://www.khanacademy.org/video/introduction-to-the-null-space-of-a-matrix Introduction to the Null Space of a Matrix]

{{linear algebra}}

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[[Category:Linear algebra]]
[[Category:Functional analysis]]
[[Category:Matrices (mathematics)]]
[[Category:Numerical linear algebra]]