Null Space · aiwiki.page

In linear algebra, the null space of a linear map is the set of vectors that it maps to the zero vector. Also called its kernel, it describes the inputs that the map annihilates. For a matrix $A$ , the null space consists of all solutions of the homogeneous equation $Ax=0$ . It is a linear subspace of the input space, not generally of the output space. (maths.tcd.ie)

Definition and basic properties

Let $T:V\to W$ be a linear map between vector spaces over a field $\mathbb F$ . Its null space is

\ker T=\{v\in V:T(v)=0_W\}.

For an $m\times n$ matrix over $\mathbb F$ , common notations include

N(A),\qquad \operatorname{Null}(A),\qquad \ker A.

Here $N(A)\subseteq\mathbb F^n$ , whereas the image of $A$ lies in $\mathbb F^m$ . (maths.tcd.ie)

The zero vector always belongs to the null space. If $u,v\in\ker T$ and $a,b\in\mathbb F$ , linearity gives

T(au+bv)=aT(u)+bT(v)=0.

Thus it is closed under linear combinations. A null space containing only the zero vector is called trivial; it is not an empty set. A linear map is an injective function precisely when its null space is trivial, because $T(u)=T(v)$ is equivalent to $u-v\in\ker T$ . (maths.tcd.ie)

Nullity and rank

The dimension of the null space is called the nullity. If $V$ is finite-dimensional, the rank–nullity theorem states

\dim V=\dim\ker T+\dim\operatorname{im}T.

Consequently, if $A$ has $n$ columns and rank $r$ ,

\operatorname{nullity}(A)=n-r.

Nullity therefore counts the independent input directions lost under the transformation. (homepages.ucl.ac.uk)

A matrix has a trivial null space exactly when its columns are linearly independent. In particular, an $m\times n$ matrix with $n>m$ necessarily has a nontrivial null space: its rank cannot exceed $m$ . This conclusion concerns homogeneous systems; it does not establish that an arbitrary equation $Ax=b$ has a solution. (live.ocw.mit.edu)

Computing a basis

For exact calculations, Gaussian elimination reduces $A$ to row-echelon form. Elementary row operations preserve the solutions of $Ax=0$ , so they preserve its null space. Pivot variables are expressed in terms of free variables. Setting one free variable to $1$ and the others to $0$ , successively, produces a basis of the null space. The number of resulting basis vectors equals the number of nonpivot columns. (ocw.mit.edu)

For example, take

A=\begin{pmatrix} 1&2&3\\ 2&4&6 \end{pmatrix}.

The second equation duplicates the first, leaving

x_1+2x_2+3x_3=0.

Writing $x_2=s$ and $x_3=t$ gives

x=s\begin{pmatrix}-2\\1\\0\end{pmatrix} +t\begin{pmatrix}-3\\0\\1\end{pmatrix}.

Hence

N(A)=\operatorname{span}\{(-2,1,0)^T,(-3,0,1)^T\}.

This calculated example has rank $1$ and nullity $2$ . Its null space is a plane through the origin in $\mathbb R^3$ , although its displayed basis is not unique.

Geometry and solution sets

For a real matrix, the null space is the orthogonal complement of the row space:

N(A)=\operatorname{im}(A^T)^\perp,

where $A^T$ denotes the transpose. Each equation in $Ax=0$ says that $x$ is perpendicular to a row of $A$ , using the standard inner product. The left null space, $N(A^T)$ , is correspondingly perpendicular to the column space and has dimension $m-r$ . For complex matrices, these statements use the conjugate transpose $A^*$ . (ocw.mit.edu)

For a consistent system of linear equations $Ax=b$ , let $x_p$ be one particular solution. Then every solution has the form

x=x_p+z,\qquad z\in N(A).

Indeed, subtracting two solutions yields a homogeneous solution, and adding any homogeneous solution preserves $Ax=b$ . The solution set is therefore an affine space obtained by translating the null space. It is a linear subspace only when $b=0$ . A consistent system has exactly one solution if and only if $N(A)=\{0\}$ . (people.cs.uchicago.edu)

Numerical computation and applications

In numerical computation, singular value decomposition can construct an orthonormal basis for the null space. In a full decomposition $A=U\Sigma V^*$ , the columns of $V$ corresponding to zero diagonal entries of $\Sigma$ , together with any additional right-side directions, span $N(A)$ . Computations with floating-point arithmetic usually classify sufficiently small singular values as zero using a tolerance. This produces an effective numerical null space, whose dimension may differ from the exact algebraic nullity. (docs.scipy.org)

In ordinary least squares, null-space components can be added to a coefficient vector without changing its fitted values. The Moore–Penrose pseudoinverse selects the least-squares solution of minimum Euclidean norm, which has no component in $N(A)$ . In a linear inverse problem, the same structure identifies changes to the unknown that cannot be distinguished by the measurements. (ocw.mit.edu)