NULL SPACE

In linear algebra, the 'null space' (also 'nullspace') of a matrix ''A'' is the set of all vectors 'x' for which ''A'''x' = '0'. The null space of a matrix with ''n'' columns is a linear subspace of ''n''-dimensional Euclidean space.
If we regard the matrix as a linear transformation, then the null space is precisely the kernel of the mapping (i.e. the set of vectors that map to zero). For this reason, the kernel of a linear transformation between abstract vector spaces is sometimes referred to as the 'null space of the transformation'.

Contents

Definition

Example

Subspace properties

Basis

Relation to the row space

Nonhomogeneous equations

Left null space

Notes

See also

References

External links

Definition

The 'null space' of an ''m'' × ''n'' matrix ''A'' is the set
:$ext\{Null\}(A)\; =\; left\{\; extbf\{x\}in\; extbf\{R\}^n\; :\; A\; extbf\{x\}\; =\; extbf\{0\}$
ight} ext{,}^[1]
where '0' denotes the zero vector with ''m'' components. The matrix equation ''A'''x' = '0' is the same as a homogeneous system of linear equations:
:
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& \
dots;;; && && dots;;; && && dots;;; && dots,& \
a_{m1} x_1 &&; + ;&& a_{m2} x_2 &&; + cdots + ;&& a_{mn} x_n &&; = 0&
end{alignat} ext{.}
From this viewpoint, the null space of ''A'' is the same as the solution set to the homogeneous system.

Example

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'':
:
2x &&; + ;&& 3y &&; + ;&& 5z &&; = ;&& 0 \
-4x &&; + ;&& 2y &&; + ;&& 3z &&; = ;&& 0
end{alignat} ext{.}
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).

Subspace properties

The null space of an ''m'' × ''n'' matrix is a subspace of 'R'^''n''. That is, the set Null(''A'') has the following three properties:
# Null(''A'') always contains the zero vector.
# If 'x' ∈ Null(''A'') and 'y' ∈ Null(''A''), then 'x' + 'y' ∈ Null(''A'').
# If 'x' ∈ Null(''A'') and ''c'' is a scalar, then ''c'''x' ∈ Null(''A'').
Here are the proofs:
# ''A'''0' = '0'.
# If ''A'''x' = '0' and ''A'''y' = '0', then ''A''('x' + 'y') = ''A'''x' + ''A'''y' = '0' + '0' = '0'.
# If ''A'''x' = '0' and ''c'' is a scalar, then ''A''(''c'''x') = ''cA'''x' = ''c'''0' = '0'.

Basis

The null space of a matrix is not affected by elementary row operations. This makes it possible to use row reduction to find a basis for the null space:
:'Input' An ''m'' × ''n'' matrix ''A''.
:'Output' A basis for the null space of ''A''
:# Use elementary row operations to put ''A'' in reduced row echelon form.
:# Interpreting the reduced row echelon form as a homogeneous linear system, determine which of the variables ''x''₁, ''x''₂, ..., ''x_n'' are free. Write equations for the dependent variables in terms of the free variables.
:# For each free variable ''x_i'', choose a vector in the null space for which ''x_i'' = 1 and the remaining free variables are zero. The resulting collection of vectors is a basis for the null space of ''A''.
;Example
:If the reduced row echelon form of ''A'' is
::
1 && 0 && -3 && 0 && 2 && 0 \
0 && 1 && 5 && 0 && -1 && 4 \
0 && 0 && 0 && 1 && 7 && -9 \
0 && ;;;;;0 && ;;;;;0 && ;;;;;0 && ;;;;;0 && ;;;;;0 end{alignat} ,
ight]
:then ''x''₃, ''x''₅, and ''x''₆ are free, with ''x''₁ = 3''x''₃ – 2''x''₅,  ''x''₂ = –5''x''₃ + ''x''₅ – 4''x''₆, and ''x''₄ = –7''x''₅ + 9''x''₆. Therefore,
::
ight],;
left[!! egin{array}{r} -2 \ 1 \ mathbf{0} \ -7 \ mathbf{1} \ mathbf{0} end{array}
ight],;
left[!! egin{array}{r} 0 \ -4 \ mathbf{0} \ 9 \ mathbf{0} \ mathbf{1} end{array}
ight]
:is a basis for the null space of ''A''.

Relation to the row space

Main articles: Rank-nullity theorem

The product of a matrix ''A'' and a vector 'x' can be written in terms of the dot product of vectors:
:
Here 'a'₁, ..., 'a'_''m'' denote the row vectors 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''.
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 null space of ''A'' is called the 'nullity' of ''A''. These quantities are related by the equation
:$ext\{rank\}(A)\; +\; ext\{nullity\}(A)\; =\; n\; ext\{.\},$
Here ''n'' denotes the number of columns of the matrix ''A''. 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:
:
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 \
dots;;; && && dots;;; && && dots;;; && &&& ;dots \
a_{m1} x_1 &&; + ;&& a_{m2} x_2 &&; + cdots + ;&& a_{mn} x_n &&; = ;&&& b_m \
end{alignat}
If 'u' and 'v' are two possible solutions to the above equation, then
:$A(\; extbf\{u\}-\; extbf\{v\})\; =\; A\; extbf\{u\}\; -\; A\; extbf\{v\}\; =\; extbf\{b\}\; -\; extbf\{b\}\; =\; extbf\{0\},$
Thus, the difference of any two solutions to the equation ''A'''x' = 'b' lies in the null space of ''A''.
It follows that any solution to the equation ''A'''x' = '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 ''A'''x' = 'b' is
:$left\{\; extbf\{v\}+\; extbf\{x\}\; ,:,\; extbf\{x\}in\; ext\{Null\}(A),$
ight} ext{,}
where 'v' is any fixed vector satisfying ''A'''v' = 'b'. Geometrically, this says that the solution set to ''A'''x' = 'b' is the translation of the null space of ''A'' by the vector 'v'.

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''.

Notes

1. This equation uses set-builder notation.

See also

★ Matrix (mathematics)

★ Euclidean subspace

★ System of linear equations

★ Row and column spaces

★ Row reduction

★ Four fundamental subspaces

References

★ Linear algebra and its applications, Lay, David C., , , Pearson/Addison-Wesley, 2006,

★ Schaum's outline of theory and problems of linear algebra, Lipson, Marc; Lipschutz, Seymour, , , McGraw-Hill, 2001,

★ Beezer, Rob, ''A First Course in Linear Algebra'', licensed under GFDL.

★ Jim Hefferon: ''Linear Algebra'' (Online textbook)

★ Edwin H. Connell: ''Elements of Abstract and Linear Algebra'' (Online textbook)

External links

★ MIT Video Lecture on Column Space and Nullspace at Google Video, from MIT OpenCourseWare