[ALGEBRA] Kernel, image and rank

The subspaces of inputs sent to zero and outputs that can occur, together with their dimensions.

Every linear map has two associated subspaces: the kernel, which holds the inputs sent to $0$ and the image, which holds the outputs that occur. This page defines both, shows they are subspaces and introduces their dimensions, nullity and rank.

Inputs that disappear

[DEFINITION]

Linear map

Let $V$ and $W$ be vector spaces over the same field $F$ . A function $T:V\to W$ is a linear map if

T(u+v)=T(u)+T(v)\qquad\text{for all }u,v\in V,

T(\omega u)=\omega T(u)\qquad\text{for all }\omega\in F,\ u\in V.

[DEFINITION]

Subspace

Let $V$ be a vector space over $F$ . A subset $S\subseteq V$ is a subspace of $V$ when it is a vector space over the same field $F$ and uses the addition and scalar multiplication inherited from $V$ . A proper subspace is a subspace that is not equal to $V$ .

[DEFINITION]

Kernel

If $T:V\to W$ is a linear map, its kernel is

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

The same set is also called the null space. If $A\in\mathbb F^{m\times n}$ is regarded as the map $\mathbb F^n\to\mathbb F^m$ , then

\ker(A)=\{x\in\mathbb F^n:Ax=0\}.

The kernel lies in the domain because it records which inputs are sent to the zero vector. For the dot-product map $T_u(x)=u\cdot x$ with $u\ne0$ , the kernel is the subspace of vectors orthogonal to $u$ .

[EXAMPLE]

Kernel of a matrix

For

A= \begin{pmatrix} 2&2&2\\ 0&3&-3 \end{pmatrix},

let $x=(x_1,x_2,x_3)^T$ . Solving $Ax=0$ gives

2x_1+2x_2+2x_3=0, \qquad 3x_2-3x_3=0.

Thus $x_2=x_3$ and $x_1=-2x_3$ . Writing $x_3=\lambda$ gives

\ker(A) = \left\{ \lambda \begin{pmatrix} -2\\1\\1 \end{pmatrix} :\lambda\in\mathbb R \right\} = \operatorname{span} \left\{ \begin{pmatrix} -2\\1\\1 \end{pmatrix} \right\}.

[routine]

[1]:Kernel

[THEOREM]

The kernel is a subspace

If $T:V\to W$ is a linear map, then $\ker(T)$ is a subspace of $V$ .

[1]:Kernel
[2]:Subspace test
[3]:Proof
[4]:Consequences of prescribed kernel and image vectors
[5]:Seen Questions

[PROOF]

Linearity gives $T(0)=0$ , so $0\in\ker(T)$ . If $u_1,u_2\in\ker(T)$ , then

T(u_1+u_2)=T(u_1)+T(u_2)=0+0=0,

so $u_1+u_2\in\ker(T)$ . If $u\in\ker(T)$ and $\lambda$ is a scalar, then

T(\lambda u)=\lambda T(u)=\lambda 0=0,

so $\lambda u\in\ker(T)$ . The subspace test now applies.

[1]:The kernel is a subspace

[DEFINITION]

Nullity

The nullity of a linear map $T$ is

\operatorname{nullity}(T)=\dim\ker(T).

For a matrix $A$ , the nullity is

\operatorname{nullity}(A)=\dim\ker(A).

Nullity counts the independent input directions that are lost completely. For a matrix, it is the dimension of the solution subspace of $Ax=0$ : the number of free parameters after Gaussian elimination and back substitution, or the number of non-pivot columns in a row-echelon form. In the kernel-of-a-matrix example above, the single spanning vector is a basis for the kernel, so $\operatorname{nullity}(A)=1$ .

[1]:Kernel
[2]:Dimension
[3]:Column independence and nullity
[4]:Rank-nullity theorem
[5]:Rank-nullity theorem for vector spaces
[6]:A projection, nullity and a composition
1 more

[PROPOSITION]

Column independence and nullity

The columns of a matrix $A$ are linearly independent if and only if $\operatorname{nullity}(A)=0$ .

A linear dependence among the columns is exactly a non-trivial solution of $Ax=0$ . Thus the columns are independent precisely when the kernel contains only the zero vector.

Outputs that can occur

[DEFINITION]

Image

If $T:V\to W$ is a linear map, its image is

\operatorname{im}(T)=\{u\in W:u=T(v)\text{ for some }v\in V\}.

The same set is also called the range. If $A\in\mathbb F^{m\times n}$ is regarded as the map $\mathbb F^n\to\mathbb F^m$ , then

\operatorname{im}(A)=\{u\in\mathbb F^m:u=Ax\text{ for some }x\in\mathbb F^n\},

which is the column space of $A$ .

The image lies in the codomain because it collects the outputs that can occur. For the dot-product map $T_u(x)=u\cdot x$ with $u\ne0$ , every real number occurs: for $\alpha\in\mathbb R$ ,

T_u\left(\frac{\alpha}{u\cdot u}u\right)=\alpha.

[1]:Linear map
[2]:Column space
[3]:Image of a matrix
[4]:The image is a subspace
[5]:Rank
[6]:Properties read from a matrix representation
1 more

[EXAMPLE]

Image of a matrix

For the matrix

A= \begin{pmatrix} 2&2&2\\ 0&3&-3 \end{pmatrix},

the image is the span of its columns. The first two columns,

\begin{pmatrix}2\\0\end{pmatrix}, \qquad \begin{pmatrix}2\\3\end{pmatrix},

are linearly independent: if $c_1(2,0)^T+c_2(2,3)^T=0$ , the second coordinate gives $c_2=0$ and then the first gives $c_1=0$ . They therefore form a basis for $\mathbb R^2$ . Hence

\operatorname{im}(A)=\mathbb R^2.

[routine]

[1]:Image

[THEOREM]

The image is a subspace

If $T:V\to W$ is a linear map, then $\operatorname{im}(T)$ is a subspace of $W$ .

[1]:Image
[2]:Subspace test
[3]:Proof
[4]:Consequences of prescribed kernel and image vectors
[5]:Seen Questions

[PROOF]

The image contains $0$ because $T(0)=0$ . If $w_1,w_2\in\operatorname{im}(T)$ , write $w_1=T(v_1)$ and $w_2=T(v_2)$ . Then

w_1+w_2=T(v_1)+T(v_2)=T(v_1+v_2),

so $w_1+w_2\in\operatorname{im}(T)$ . If $w=T(v)$ , then

\lambda w=\lambda T(v)=T(\lambda v),

so $\lambda w\in\operatorname{im}(T)$ . The subspace test now applies.

[1]:The image is a subspace

[DEFINITION]

Rank

The rank of a linear map $T$ is

\operatorname{rank}(T)=\dim\operatorname{im}(T).

For a matrix $A$ , the rank is

\operatorname{rank}(A)=\dim\operatorname{im}(A).

For a matrix, rank is also the maximum number of linearly independent columns, or the number of pivot columns in a row-echelon form. In the image-of-a-matrix example above, the two displayed columns form a basis of the image, so $\operatorname{rank}(A)=2$ .

“Pivot columns” refers to positions in the echelon form. When row operations have been used, take the columns in those positions from the original matrix to obtain a basis for the image.

[1]:Image
[2]:Dimension
[3]:Rank-nullity theorem
[4]:Rank-nullity theorem for vector spaces
[5]:Kernels, ranks and powers of a matrix
[6]:Ranks of products involving a matrix and its transpose
1 more

The kernel and image can lie in different spaces: for $T:V\to W$ , the kernel is a subspace of the domain $V$ and the image is a subspace of the codomain $W$ . Nullity measures the first and rank the second. Rank-nullity relates both to the dimension of the domain.

Kernel, image and rank

Linear map

Subspace

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