Kernel and image - Ciallo Wiki

1 Kernel (Null Space)

1.1 Definition and basic properties

For a linear map \(T: V \to W\), the kernel (or null space) is the set \[ \ker(T)=\{v\in V \mid T(v)=0\}, \] where \(0\) denotes the zero vector in \(W\). The kernel collects exactly those inputs that the map annihilates. It is central because it measures “loss of information”: distinct vectors in \(V\) that differ by an element of \(\ker(T)\) can produce the same output.

Basic properties follow immediately from linearity. In particular, if \(v_1, v_2 \in \ker(T)\), then \(T(v_1)=0\) and \(T(v_2)=0\), so \[ T(v_1+v_2)=T(v_1)+T(v_2)=0,\quad T(cv_1)=cT(v_1)=0, \] for any scalar \(c\). Thus the kernel behaves like a linear object, not just a set.

1.2 Computing the kernel

1.2.1 Homogeneous systems and row reduction

When \(T\) is represented by a matrix \(A\) with respect to chosen bases, the kernel corresponds to solutions of a homogeneous system: \[ A x = 0. \] Computationally, the standard approach uses row reduction (Gaussian elimination) to transform \(A\) into a reduced row echelon form. The pivot/non-pivot structure reveals constraints on the variables, and therefore determines which vectors \(x\) satisfy the equation.

Concretely, after row reduction, each free variable can be chosen freely, and the pivot variables are expressed in terms of these choices. The resulting parameterized solution set is precisely \(\ker(A)\), and hence \(\ker(T)\).

1.2.2 Parameterizations of solution spaces

Because \(\ker(T)\) is a vector subspace, solutions of \(A x=0\) can be written as a linear combination of basis vectors for the null space. A typical workflow is:

Find a basis \(\{z_1,\dots,z_k\}\) for the solution space using the reduced system.
Express every kernel vector as

\[ x = \alpha_1 z_1 + \cdots + \alpha_k z_k. \] This parameterization makes both computation and theoretical reasoning easier—for instance, it quickly supports checks of membership in the kernel and helps compare kernels of related maps.

1.3 Kernel as a subspace

1.3.1 Dimension of the kernel (nullity)

The kernel is a subspace of \(V\). Its dimension is called the nullity of \(T\): \[ \operatorname{nullity}(T)=\dim(\ker(T)). \] Nullity quantifies how many degrees of freedom remain once the map’s constraints are imposed. A linear map is injective precisely when its kernel contains only the zero vector, which corresponds to nullity \(0\).

1.4 Relationship to injectivity

For a linear map \(T: V \to W\), the following are equivalent:

\(T\) is injective.
\(\ker(T)=\{0\}\).
\(\operatorname{nullity}(T)=0\).

Intuitively, injectivity means no nonzero input is sent to the same output as another input. Since \(T(v)=0\) characterizes those inputs that collapse to the zero output, having only the zero vector in the kernel eliminates ambiguity and guarantees uniqueness.

1.5 Kernel of compositions and restrictions

Two related operations frequently used in linear algebra are composing maps and restricting a map to a subspace.

Kernel of a composition. For linear maps \(S: U\to V\) and \(T: V\to W\), consider \(T\circ S: U\to W\). If \(u\in \ker(T\circ S)\), then \[ (T\circ S)(u)=0 \quad \Longleftrightarrow\quad S(u)\in \ker(T). \] Thus \(\ker(T\circ S)\) consists of those \(u\) whose image under \(S\) lands in \(\ker(T)\). This gives a useful inclusion relationship that supports computations in multi-step transformations.

Kernel under restriction. If \(U'\subseteq V\) and one considers the restricted map \(T	_{U'}: U' \to W\), then

\ker(T	_{U'}) = U' \cap \ker(T).

\] So restriction cannot enlarge the kernel; it selects those kernel vectors that lie inside the chosen domain subspace.

2 Image (Range)

2.1 Definition and basic properties

For \(T: V \to W\), the image (or range) is \[ \operatorname{im}(T)=\{T(v)\mid v\in V\}. \] Unlike the kernel, which captures inputs that map to zero, the image captures which outputs are actually attainable. It is the “effective output space” of the transformation.

Linearity implies that \(\operatorname{im}(T)\) is closed under addition and scalar multiplication. If \(y_1=T(v_1)\) and \(y_2=T(v_2)\), then \[ y_1+y_2 = T(v_1)+T(v_2)=T(v_1+v_2)\in \operatorname{im}(T), \] and for scalar \(c\), \[ cy_1=cT(v_1)=T(cv_1)\in \operatorname{im}(T). \] Therefore the image is a subspace of \(W\).

2.2 Computing the image

2.2.1 Column space interpretation

When \(T\) is represented by a matrix \(A\) (with domain basis vectors corresponding to columns), the image is the column space of \(A\). If \(A\) has columns \(a_1,\dots,a_n\), then \[ Ax = x_1 a_1 + \cdots + x_n a_n, \] so every output vector is a linear combination of the columns. Consequently, \[ \operatorname{im}(T) = \operatorname{col}(A) = \operatorname{span}\{a_1,\dots,a_n\}. \] This interpretation aligns computational methods with familiar geometric and spanning-set reasoning.

2.2.2 Spanning sets and basis extraction

To compute \(\operatorname{im}(T)\) concretely, one typically:

Identify a spanning set for the output (often the columns of \(A\)).
Reduce redundancy by extracting a basis.

In practice, row reduction or column operations are used to find which columns are linearly independent. The independent columns form a basis of the column space. The span of these selected columns then equals \(\operatorname{im}(T)\).

An equivalent approach is to compute \(\operatorname{im}(T)\) from a basis of \(V\): apply \(T\) to each basis vector to get candidate output vectors, then reduce them to a basis of their span.

2.3 Image as a subspace

As with the kernel, the image is a subspace of the codomain. This subspace structure is crucial for applying dimension arguments, orthogonal complement relationships (when an inner product exists), and induced-map constructions on quotients.

2.4 Dimension of the image (rank)

The dimension of the image is called the rank of \(T\): \[ \operatorname{rank}(T)=\dim(\operatorname{im}(T)). \] If \(T\) is represented by a matrix \(A\), then \(\operatorname{rank}(T)\) equals the rank of \(A\), meaning the number of pivot columns (equivalently, the number of pivot rows in row-echelon form).

Rank measures how many independent directions in \(W\) can be produced by the map, reflecting the “output capacity” of \(T\).

2.5 Relationship to surjectivity

Surjectivity means every vector in \(W\) is achieved as an output: \[ \operatorname{im}(T)=W. \] This is equivalent to having full rank: \[ \operatorname{rank}(T)=\dim(W), \] when \(W\) is finite-dimensional. Thus the image provides a direct test of whether the map hits the entire codomain.

In finite-dimensional settings, surjectivity and injectivity are tied together by dimension: if \(\dim(V)=\dim(W)\), then \(T\) is injective iff it is surjective, which appears naturally when rank–nullity is applied.

3 Kernel–Image Theorem and Rank–Nullity

3.1 Rank–nullity theorem statement

For a linear map \(T: V \to W\) between finite-dimensional vector spaces, \[ \dim(V)=\operatorname{rank}(T)+\operatorname{nullity}(T). \] Since \(\operatorname{nullity}(T)=\dim(\ker(T))\) and \(\operatorname{rank}(T)=\dim(\operatorname{im}(T))\), the theorem can also be written as \[ \dim(V)=\dim(\operatorname{im}(T))+\dim(\ker(T)). \] This identity links the internal collapse caused by the kernel with the external reach described by the image.

3.2 Consequences and interpretive viewpoints

Several practical consequences follow:

Bounding rank and nullity. Knowing one determines the other through the dimension of the domain.
Detecting injectivity and surjectivity. As noted, injectivity corresponds to nullity \(0\), while surjectivity corresponds to rank equal to \(\dim(W)\).
Structured decomposition. The theorem supports the idea that \(V\) splits into a part that controls outputs (related to a complement of the kernel) and a part that maps to zero (the kernel itself).

Interpretively, rank quantifies the degrees of freedom that survive transformation, while nullity quantifies degrees of freedom that are “forgotten” by mapping to zero.

3.3 Examples with full computation

Example. Let \(T:\mathbb{R}^3\to\mathbb{R}^2\) be represented by the matrix \[ A=\begin{pmatrix} 1 & 2 & 3\\ 2 & 4 & 6 \end{pmatrix}. \]

Compute the kernel. Solve \(Ax=0\) for \(x=(x_1,x_2,x_3)^T\):

\[ \begin{pmatrix} 1 & 2 & 3\\ 2 & 4 & 6 \end{pmatrix} \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix} = \begin{pmatrix}x_1+2x_2+3x_3\\2x_1+4x_2+6x_3\end{pmatrix} =\begin{pmatrix}0\\0\end{pmatrix}. \] The second equation is twice the first, so there is one independent constraint: \[ x_1+2x_2+3x_3=0. \] Let \(x_2=s\) and \(x_3=t\). Then \(x_1=-2s-3t\), so \[ x = \begin{pmatrix}-2s-3t\\ s\\ t\end{pmatrix} = s\begin{pmatrix}-2\\1\\0\end{pmatrix}+t\begin{pmatrix}-3\\0\\1\end{pmatrix}. \] Hence \(\dim(\ker(T))=2\), so \(\operatorname{nullity}(T)=2\).

Compute the image. The image is the column space of \(A\). Observe that the second column is \(2\) times the first, and the third is \(3\) times the first:

\[ \begin{pmatrix}2\\4\end{pmatrix}=2\begin{pmatrix}1\\2\end{pmatrix},\quad \begin{pmatrix}3\\6\end{pmatrix}=3\begin{pmatrix}1\\2\end{pmatrix}. \] Therefore the columns span a one-dimensional subspace generated by \(\begin{pmatrix}1\\2\end{pmatrix}\). Thus \(\dim(\operatorname{im}(T))=1\), so \(\operatorname{rank}(T)=1\).

Check rank–nullity. Here \(\dim(V)=3\), \(\operatorname{rank}(T)=1\), and \(\operatorname{nullity}(T)=2\). Indeed,

\[ 3=1+2. \]

3.4 Graphical and geometric intuition

Geometrically, the kernel is the set of vectors mapped to the origin, forming a “directional” subspace where the transformation collapses. The image is the set of all outputs obtainable by moving inputs through the domain, forming a subspace in the codomain.

In finite-dimensional Euclidean spaces, one may visualize a linear transformation as flattening or bending space so that certain directions contribute to movement in \(W\) (captured by the image), while other directions become invisible because they map to zero (captured by the kernel). The rank–nullity theorem then states that the number of independent input directions equals the number of independent output directions plus the number of independent “collapsed” directions.

4 Structural Connections

4.1 Quotient spaces and induced maps

A quotient space identifies vectors that differ by an element of a subspace. Specifically, for \(\ker(T)\subseteq V\), vectors in the same coset \(v+\ker(T)\) behave identically under \(T\) because \[ T(v+k)=T(v)+T(k)=T(v)+0=T(v) \] for \(k\in\ker(T)\).

This allows defining an induced map from the quotient: \[ \tilde{T}: V/\ker(T)\to \operatorname{im}(T), \] which is well-defined and captures the “essential action” of \(T\). In this formulation, all kernel directions are collapsed at the level of the domain, and the transformation becomes injective onto its image.

4.2 Factoring a linear map through its image

A linear map can be decomposed into simpler pieces. There is a standard factorization: \[ V \xrightarrow{\pi} V/\ker(T) \xrightarrow{\tilde{T}} \operatorname{im}(T) \hookrightarrow W, \] where \(\pi\) is the quotient projection and \(\tilde{T}\) is the induced map described above. The final inclusion \(\operatorname{im}(T)\hookrightarrow W\) simply views the image as a subspace of the codomain.

This factorization clarifies the role of kernel and image: the kernel determines what must be identified in the domain, while the image records the outcomes that survive. Many theoretical results become more transparent when phrased in terms of this structure.

4.3 Orthogonal complements (if using inner products)

When \(V\) and \(W\) carry inner products (or at least when an inner-product structure is available to define orthogonality), additional relationships connect kernels and images via the adjoint map \(T^*\).

A common identity is: \[ (\operatorname{im}(T))^\perp = \ker(T^*), \quad\text{and}\quad (\ker(T))^\perp = \operatorname{im}(T^*). \] These statements link “what is orthogonal to all outputs” with “inputs that become zero under the adjoint.” They support computational strategies and conceptual proofs, especially in least-squares and projection problems.

4.4 Computing kernels/images via eigen and invariant subspaces

For certain classes of maps, especially operators on a single vector space, eigenstructure and invariant subspaces can simplify kernel and image computations.

If \(T: V\to V\) has eigenvalue \(0\), then the corresponding eigenspace satisfies \[ E_0=\{v\in V\mid T(v)=0\}=\ker(T), \] so the kernel can be read directly from eigenvectors associated with eigenvalue zero.

More generally, if a subspace \(U\subseteq V\) is invariant under \(T\) (meaning \(T(U)\subseteq U\)), then kernel and image restricted to \(U\) can be analyzed within \(U\). For example, the kernel of the restriction \(T	_U\) is \(\ker(T	_U)=U\cap\ker(T)\), and the image of the restriction lies in \(U\). Decomposing \(V\) into invariant components (as in Jordan or spectral decompositions, when applicable) can turn a global computation into smaller, structured problems.