Pre-Images of a Linear Transformation

Overview

A function guarantees one output for each input, but says nothing in the other direction. A given output may arise from many inputs, from exactly one, or from none at all. The pre-image of a vector collects all the inputs that produce it, and for a linear transformation this set has a rigid structure that is worth knowing before any computation begins.

Two questions are answered at once. Whether the pre-image is empty asks whether the target output is achievable — a reachability, consistency or feasibility question. How large it is asks whether the input that produced an observed output can be identified uniquely — an observability, identifiability or redundancy question. In matrix terms these are the consistency of $A\vec{x} = \vec{b}$ and the dimension of the null space of $A$ , so a single computation settles both.

The structural result is that a non-empty pre-image is never a subspace unless the target is the zero vector, but it is always a translate of one. Find any single input $\vec{u}_p$ with $T(\vec{u}_p) = \vec{v}$ , and the complete pre-image is $\vec{u}_p$ plus the kernel, the set of inputs mapped to zero. This is the same particular-plus-homogeneous decomposition that describes the solution set of a linear system, restated for functions, and it means every non-empty pre-image has the same shape and the same dimension.

The engineering value lies in that shared shape. When a system has more input channels than the output constrains, every achievable target is produced by an entire family of commands differing by elements of the kernel. Those differences cost nothing in output, so they are free to be spent on a secondary objective — minimum energy, joint limit avoidance, actuator wear balancing. Redundancy resolution in robotics, null-space control and minimum-norm allocation are all the same observation applied to a pre-image.

Definition

Pre-Image

For a linear transformation $T\colon U \to V$ and a vector $\vec{v} \in V$ , the pre-image of $\vec{v}$ is the subset of the domain $T^{-1}(\vec{v}) = \{\, \vec{u} \in U \mid T(\vec{u}) = \vec{v} \,\}$ . It contains every input that $T$ sends to $\vec{v}$ , and it may be empty.

The notation $T^{-1}$ here denotes a set-valued operation and carries no implication that $T$ is invertible. Every linear transformation has pre-images; only bijective ones have inverse functions.

Kernel

KLT

The pre-image of the zero vector, $\mathcal{K}(T) = T^{-1}(\vec{0}) = \{\, \vec{u} \in U \mid T(\vec{u}) = \vec{0} \,\}$ . Unlike a general pre-image it is a subspace of $U$ : it contains $\vec{0}$ because $T(\vec{0}) = \vec{0}$ , and it is closed under addition and scalar multiplication because $T$ is linear. For a transformation represented by a matrix $A$ , the kernel is the null space $\mathcal{N}(A)$ .

Range

RLT

The set of vectors actually produced by $T$ , written $\mathcal{R}(T) = \{\, T(\vec{u}) \mid \vec{u} \in U \,\} \subseteq V$ . A pre-image is non-empty precisely when the target lies in the range. For a matrix representation, the range is the column space $\mathcal{C}(A)$ , so reachability is decided by row reduction of the augmented array.

Coset of a Subspace

CST

A set of the form $\vec{u}_p + W = \{\, \vec{u}_p + \vec{w} \mid \vec{w} \in W \,\}$ , where $W$ is a subspace and $\vec{u}_p$ a fixed vector. A coset is a subspace translated away from the origin; it is itself a subspace only when $\vec{u}_p \in W$ , equivalently when the coset contains $\vec{0}$ . Every non-empty pre-image is a coset of the kernel.

Concepts

Emptiness is a reachability question

The pre-image $T^{-1}(\vec{v})$ is non-empty if and only if $\vec{v}$ belongs to the range of $T$ . For a transformation given by a matrix, this is exactly the consistency of the system $A\vec{x} = \vec{v}$ , decided by row-reducing the augmented matrix $[\, A \mid \vec{v} \,]$ and checking whether the final column contains a leading one. An empty pre-image is not a failure of computation; it is the correct answer to the question of whether a target output can be produced at all.

A non-empty pre-image is a translate of the kernel

Suppose $T(\vec{u}_p) = \vec{v}$ . If $\vec{u}$ is any other element of the pre-image then $T(\vec{u} - \vec{u}_p) = T(\vec{u}) - T(\vec{u}_p) = \vec{v} - \vec{v} = \vec{0}$ , so their difference lies in the kernel. Conversely, adding any kernel element to $\vec{u}_p$ leaves the output unchanged. Hence $T^{-1}(\vec{v}) = \vec{u}_p + \mathcal{K}(T)$ exactly. This is the particular-plus-homogeneous structure of linear solution sets, and it says that every non-empty pre-image is a shifted copy of one fixed subspace.

A pre-image is almost never a subspace

Because $T(\vec{0}) = \vec{0}$ , the zero vector belongs to $T^{-1}(\vec{v})$ only when $\vec{v} = \vec{0}$ . A subspace must contain the zero vector, so for $\vec{v} \ne \vec{0}$ the pre-image fails the very first test. It is also not closed: the sum of two elements of $T^{-1}(\vec{v})$ maps to $2\vec{v}$ , not to $\vec{v}$ . The only pre-image that is a subspace is the kernel itself. This is the single most common misconception about pre-images and it matters, because subspace machinery — bases, dimension, spans — applies to the kernel but not to the shifted copy.

Pre-images partition the domain

Every vector of $U$ lies in exactly one pre-image, namely $T^{-1}(T(\vec{u}))$ , and two pre-images of different targets are disjoint. The non-empty pre-images therefore form a partition of the domain, the equivalence classes of the relation "produces the same output". Each class is a translate of the kernel, so all classes have the same dimension, and the number of distinct classes is measured by the dimension of the range. This decomposition is what the rank-nullity theorem counts.

Pre-image sizes characterise injectivity and surjectivity

The transformation is injective exactly when no pre-image contains two distinct elements, which happens exactly when the kernel is trivial — the translate then collapses to a single point. It is surjective exactly when no pre-image is empty, that is when the range fills the codomain. It is bijective when every pre-image contains exactly one element, at which point $T^{-1}$ genuinely is a function and the notation loses its set-valued reading. Framing all three properties in terms of pre-image sizes makes their relationship immediate.

Computing a pre-image in practice

For $T(\vec{x}) = A\vec{x}$ , row-reduce $[\, A \mid \vec{v} \,]$ . A leading one in the final column reports an empty pre-image and the work stops. Otherwise, set the free variables to zero to read a particular solution $\vec{u}_p$ , and read a basis for the null space from the same reduced form. The answer is $\vec{u}_p + \operatorname{span}\{\, \vec{z}_1, \ldots, \vec{z}_k \,\}$ with $k = \operatorname{nullity}(A)$ . One reduction produces the consistency verdict, the particular solution and the kernel basis together.

Choosing a representative from the pre-image

When the pre-image has positive dimension, every element is an equally valid answer to the reachability question, so a secondary criterion must decide which one to use. The minimum Euclidean norm element is the standard default: it is the unique point of the coset orthogonal to the kernel, and it is what the pseudoinverse returns. Other criteria — staying within actuator limits, minimising a weighted energy, maximising distance from a singular configuration — are formulated as optimisation over the coset, which is a linearly constrained problem in $k$ free parameters rather than in $n$ .

Procedure: determine a pre-image

Write the defining conditionSet

T(\vec{u}) = \vec{v}

with

\vec{u}

a generic element of the domain, and expand both sides. For abstract spaces, equating components or entries turns this into a linear system.

Row-reduce the augmented arrayReduce

[\, A \mid \vec{v} \,]

where

A

is the matrix representation. This single reduction answers every question that follows.

Test the final column for a pivotA leading one in the augmented column means the system is inconsistent, so

T^{-1}(\vec{v}) = \emptyset

and

\vec{v}

is outside the range. Stop here.

Extract a particular elementSet every free variable to zero and read the dependent variables from the reduced rows. That vector

\vec{u}_p

is one valid input producing

\vec{v}

Extract a basis for the kernelFrom the same reduced form, read the null space basis: one vector per free variable. Its dimension is

\operatorname{nullity}(A) = n - \operatorname{rank}(A)

Assemble and, if needed, optimiseReport

T^{-1}(\vec{v}) = \vec{u}_p + \operatorname{span}\{\vec{z}_i\}

. If a single command is required, choose the element minimising norm, energy or actuator excursion over the free parameters.

Equations

Definition of the pre-image

EQ-PI-01

Definition of the pre-image

A subset of the domain, possibly empty. The superscript $-1$ is set notation and does not assert that $T$ has an inverse function.

Non-emptiness criterion

EQ-PI-02

Non-emptiness criterion

For a matrix representation this reads $\vec{v} \in \mathcal{C}(A)$ , decided by the consistency of $A\vec{x} = \vec{v}$ .

Coset structure of a non-empty pre-image

EQ-PI-03

Coset structure of a non-empty pre-image

One particular input plus every input mapped to zero. All non-empty pre-images are translates of the same subspace and therefore have the same dimension.

The kernel is the pre-image of zero

EQ-PI-04

K (T) = T^{- 1} (\vec{0}) = N (A)

This is the one pre-image that is a subspace of the domain, and the one whose basis is needed to describe all the others.

Difference of two elements lies in the kernel

EQ-PI-05

{\vec{u}}_{1}, {\vec{u}}_{2} \in T^{- 1} (\vec{v}) \Rightarrow T ({\vec{u}}_{1} - {\vec{u}}_{2}) = \vec{0}

The proof of the coset structure in one line, and the practical statement that two commands achieving the same output differ by a null-space motion.

Pre-images partition the domain

EQ-PI-06

Pre-images partition the domain

Every input belongs to exactly one pre-image. The classes all have dimension $\operatorname{nullity}$ , and the range indexes them — the geometric content of the rank-nullity theorem.

Pre-image sizes classify the transformation

EQ-PI-07

Pre-image sizes classify the transformation

Injectivity is the trivial-kernel condition; surjectivity is the full-range condition. Both hold together exactly when every pre-image is a single point.

Minimum-norm element of a pre-image

EQ-PI-08

Minimum-norm element of a pre-image

The unique element of the coset orthogonal to the kernel. It is what the Moore-Penrose pseudoinverse returns and is the standard default when a single command must be chosen.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$T$	Linear transformation	The map whose pre-images are being studied	linear map from U to V
$\vec{v}$	Target vector	The codomain element whose pre-image is required	element of V
$Pre-image$	Pre-image	Set of all inputs mapped to $\vec{v}$ ; possibly empty	subset of U
${\vec{u}}_{p}$	Particular element	Any single input with $T(\vec{u}_p) = \vec{v}$	element of U
$K (T)$	Kernel	Subspace of inputs mapped to the zero vector	subspace of U
$R (T)$	Range	Set of outputs actually produced; equals $\mathcal{C}(A)$ for a matrix representation	subspace of V
${\vec{z}}_{i}$	Kernel basis vector	One of the $\operatorname{nullity}$ directions along which the output does not change	element of K(T)
$k$	Nullity	Dimension of the kernel, hence of every non-empty pre-image	integer 0 to n
$A$	Matrix representation	Matrix with $T(\vec{x}) = A\vec{x}$ , used to compute pre-images by row reduction	m x n matrix

Worked Numerical Example

Problem statement

A positioning stage carries three actuators. Each unit of actuator command contributes a fixed pattern to four monitored channels — two force components, a moment and a strain reading. Determine which commands achieve a specified monitored pattern, whether a second pattern is achievable at all, and which achieving command minimises actuator effort.

State the transformation
The command vector has three components and the monitored pattern four, so $T\colon \mathbb{C}^3 \to \mathbb{C}^4$ with the matrix below. Note that column three is the sum of columns one and two: actuator three does exactly what actuators one and two do together.
$A = [\begin{matrix} 1 & 0 & 1 \\ 2 & 1 & 3 \\ 3 & 1 & 4 \\ 1 & - 1 & 0 \end{matrix}], {\vec{A}}_{3} = {\vec{A}}_{1} + {\vec{A}}_{2}$
Set up the pre-image of the target pattern
The commissioning target is the monitored pattern $\vec{v} = (2, 5, 7, 1)^{t}$ . The pre-image is the solution set of $A\vec{u} = \vec{v}$ , so form the augmented array and reduce it.
$[\begin{matrix} 1 & 0 & 1 & 2 \\ 2 & 1 & 3 & 5 \\ 3 & 1 & 4 & 7 \\ 1 & - 1 & 0 & 1 \end{matrix}] ⟶ [\begin{matrix} 1 & 0 & 1 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$
Read the consistency verdict and the structure
The final column contains no leading one, so the pre-image is non-empty and the target is achievable. There are two pivot columns, giving $\operatorname{rank}(A) = 2$ , and one free variable $u_3$ , giving $\operatorname{nullity}(A) = 1$ . The pre-image is therefore a line in command space, not a point.
Extract a particular command
Set the free variable to zero. The reduced rows then read $u_1 = 2$ and $u_2 = 1$ directly. Verify by substitution: $2\vec{A}_1 + 1\vec{A}_2 = (2,4,6,2)^{t} + (0,1,1,-1)^{t} = (2,5,7,1)^{t}$ , which is the target.
${\vec{u}}_{p} = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}], A {\vec{u}}_{p} = [\begin{matrix} 2 \\ 5 \\ 7 \\ 1 \end{matrix}] = \vec{v}$
Extract the kernel and assemble the pre-image
With $u_3$ free, the reduced rows give $u_1 = -u_3$ and $u_2 = -u_3$ for the homogeneous system. The kernel is spanned by $(-1, -1, 1)^{t}$ , which is the null motion: driving actuators one and two backwards while driving actuator three forwards produces no change in any monitored channel.
$T^{- 1} (\vec{v}) = [\begin{matrix} 2 \\ 1 \\ 0 \end{matrix}] + u_{3} [\begin{matrix} - 1 \\ - 1 \\ 1 \end{matrix}], u_{3} \in ℂ$
Test a second, unreachable target
Now take $\vec{w} = (1, 0, 0, 0)^{t}$ — a unit force in the first channel with everything else held at zero. Reducing the corresponding augmented array produces a leading one in the final column, the signature of an inconsistent system.
$[\begin{matrix} 1 & 0 & 1 & 1 \\ 2 & 1 & 3 & 0 \\ 3 & 1 & 4 & 0 \\ 1 & - 1 & 0 & 0 \end{matrix}] ⟶ [\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}], T^{- 1} (\vec{w}) = \emptyset$
Choose the minimum-effort command
Every point on the line achieves the target equally well, so the free parameter is available for a secondary objective. Minimising the squared Euclidean command norm gives a quadratic in $u_3$ with its minimum at $u_3 = 1$ .
$Choose the minimum-effort command$
Report the selected command
Substituting $u_3 = 1$ gives the minimum-norm element, with squared norm $2$ against $5$ for the naive particular solution — a reduction in command magnitude of about $37\%$ for identical monitored output. Confirm it is orthogonal to the kernel: $\langle (1,0,1)^{t}, (-1,-1,1)^{t} \rangle = -1 + 0 + 1 = 0$ .
${\vec{u}}_{min} = [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}], ‖ {\vec{u}}_{min} ‖ = \sqrt{2} against ‖ {\vec{u}}_{p} ‖ = \sqrt{5}$

Result

The commissioning target is achievable by a one-parameter family of commands, and choosing the minimum-norm member cuts the command magnitude by more than a third with no change in monitored output. The second target is unreachable: no combination of the three actuators produces a pure first-channel force, because the achievable patterns fill only a two-dimensional subspace of the four monitored channels. The redundancy and the unreachability come from the same rank deficiency — actuator three duplicates the combined action of the other two — and both are read from a single row reduction.

Applications & Industry Use

Robotics

Redundancy resolution and null-space motion

For a manipulator with more joints than task dimensions, the pre-image of a desired end-effector velocity is a coset of the Jacobian null space. Motions within that null space change the joint configuration while leaving the end-effector still, which is exploited to avoid joint limits, keep away from singular configurations and minimise energy without disturbing the task.

Control allocation

Distributing a command among redundant actuators

Aircraft with multiple control surfaces, ships with several thrusters and electric vehicles with independent wheel motors all face the problem of realising a commanded generalised force with more actuators than degrees of freedom. The set of admissible allocations is a pre-image, and the allocation law picks a member subject to rate limits, saturation and wear balancing.

Chemical engineering

Feasible feed combinations for a target product

Asking which feed rates produce a specified product composition is a pre-image query. An empty pre-image means the specification cannot be met with the available feeds — information that should stop an optimisation before it starts — and a positive-dimensional pre-image identifies the degrees of freedom left for cost minimisation.

Medical imaging & tomography

Non-uniqueness in reconstruction

A limited-angle or under-sampled acquisition has a non-trivial null space, so many images are consistent with the same measurements. The pre-image of the measured data is the full set of candidate reconstructions, and regularisation is precisely the rule that selects one member of it — minimum norm, minimum total variation, or maximum likelihood under a prior.

Structural engineering

Self-stress states in indeterminate structures

The pre-image of the zero load vector under the equilibrium operator is the set of internal force states requiring no external load: the self-stress states. Their dimension is the degree of static indeterminacy, and they are what allow pre-tensioning of cable nets and post-tensioning of concrete without disturbing external equilibrium.

Cryptography & coding theory

Syndrome decoding

For a linear code, the pre-image of a computed syndrome under the parity check map is the coset of error patterns consistent with the received word. Decoding chooses the minimum-weight member of that coset, which is the coding-theoretic analogue of choosing the minimum-norm element of a pre-image.

Design Considerations

Answer the emptiness question before anything else

A great deal of computation is wasted attempting to realise unreachable targets. One row reduction of the augmented array returns the verdict, and an empty pre-image is a design finding, not an error: it says the actuator set spans too small a subspace and no control law can fix that.

Do not treat a pre-image as a subspace

Only the pre-image of the zero vector is a subspace. Attempting to take a basis of a general pre-image, or to add two of its elements, produces nonsense. Handle it as a fixed offset plus a subspace, and confine all subspace operations to the kernel part.

Choose the representative deliberately

When the pre-image has positive dimension, every element is a correct answer and the software will return whichever one the algorithm happens to produce. Minimum Euclidean norm is a defensible default and is what a pseudoinverse gives, but it takes no account of actuator limits or differing actuator costs. State the selection rule explicitly as part of the design.

Nullity is the size of the design freedom

The dimension of the kernel counts the free parameters available for secondary objectives. A nullity of zero means the command is uniquely determined and no allocation freedom exists; a large nullity means most of the input space is invisible at the output, which is useful for optimisation and dangerous for diagnosis, since faults occurring within the kernel produce no observable symptom.

Rank is a tolerance decision on measured operators

An identified or measured matrix rarely has an exactly rank-deficient structure; instead it has small singular values. The pre-image is then formally a single point but practically an extremely elongated set, and a naive solve produces a huge command. Use a truncated singular value decomposition or regularisation with an explicit threshold rather than an exact-zero pivot test.

Optimise over the coset, not over the whole domain

Selecting the best element of a pre-image is a constrained problem in $n$ variables, but parametrising the coset as $\vec{u}_p + \sum_i t_i \vec{z}_i$ turns it into an unconstrained problem in $k = \operatorname{nullity}$ variables. For a quadratic objective this reduces to a small symmetric linear system, which is far cheaper and better conditioned than the constrained formulation.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Specifies set-builder and function notation, including the convention that $T^{-1}$ applied to a set or element denotes a pre-image rather than an inverse function.
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Provides the minimum-norm least-squares driver `xGELSD`, which returns the minimum-norm element of a pre-image and reports the effective rank against a supplied tolerance.
`ISO 9283`	Manipulating industrial robots — Performance criteria and related test methods	Defines pose accuracy and repeatability measures for manipulators, the metrics against which a redundancy resolution scheme selecting an element of a pre-image must be assessed.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Governs the arithmetic in which a structurally rank-deficient operator appears merely nearly deficient, which is why pre-image dimension must be determined against a stated tolerance.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the set-builder notation of the pre-image definition semantically, keeping the conditional bar and the membership symbol distinguishable to assistive technology.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact rational row reduction	Small analytic systems where the exact dimension of the pre-image and an exact kernel basis are required.	Gives an unambiguous consistency verdict and exact structure, but is unavailable for measured operators and impractical beyond modest sizes.
IEEE 754 binary64 with LU and a pivot tolerance	Well-conditioned square or overdetermined systems where the operator is known to have full rank.	Fastest route to a particular element, but gives no reliable information about a near-deficient kernel and will silently return a huge command near singularity.
Truncated singular value decomposition	Measured or identified operators where rank must be judged against noise, as in tomography and system identification.	Yields the minimum-norm element and an explicit, defensible rank threshold, at $O(mn^2)$ cost and with the threshold itself becoming a design parameter that must be justified.
Moore-Penrose pseudoinverse	A single default representative of every pre-image is needed and minimum norm is an acceptable criterion.	Uniform and well defined for any operator, but the minimum-norm choice ignores actuator limits and differing actuator costs, so it is rarely the right engineering answer without weighting.
Weighted or constrained allocation solve	Actuators have unequal authority, cost or saturation limits, as in control allocation for aircraft and marine vessels.	Selects an operationally sensible element of the pre-image, but requires a quadratic or linear programme per sample and needs a fallback when the constrained problem is infeasible.
Sparse factorisation with fill-reducing ordering	Very large structural or network operators where the pre-image is needed but the matrix cannot be treated as dense.	Preserves sparsity and makes the problem tractable, but recovering a full kernel basis remains expensive, so the null space is usually characterised by a few iteratively computed vectors instead.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

One reduction answers everything

Row-reducing $[\, A \mid \vec{v} \,]$ at a cost of about $\tfrac{2}{3}n^3$ operations returns the consistency verdict, a particular element and a kernel basis simultaneously. Performing separate computations for reachability, for a particular solution and for the null space triples the work and risks the three results being inconsistent with one another.

Doing it by hand

Reduce the augmented array fully, check the final column for a leading one, then set every free variable to zero to read the particular element. For the kernel, set one free variable to $1$ and the rest to zero in turn, reading one basis vector each time. Verify the particular element by substitution into the original transformation, not into the reduced system, since an arithmetic slip during reduction is invisible to the latter check.

Library behaviour and the minimum-norm default

numpy.linalg.lstsq and scipy.linalg.lstsq return the minimum-norm element of the pre-image for rank-deficient input, along with the effective rank determined against a tolerance parameter. numpy.linalg.solve raises an error rather than returning a representative, and scipy.linalg.null_space supplies a kernel basis from a singular value decomposition. Read the documented tolerance before trusting a reported rank.

Verification that catches real errors

Two checks are worth the effort. Substitute the particular element into the original transformation and compare with the target. Then apply the transformation to each kernel basis vector and confirm the result is the zero vector to working precision. Together these certify both halves of the coset description independently of the reduction that produced them.

Cost of selecting a representative

Once a kernel basis is available, minimising a quadratic objective over the coset reduces to a $k \times k$ symmetric system, where $k$ is the nullity — usually a very small problem. Recomputing the whole pre-image for each new target is unnecessary when only the target changes: the kernel is fixed, so only the particular element must be recomputed, at the cost of a triangular solve if the factorisation is retained.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Treating a pre-image as a subspace	high	Assuming that because the kernel is a subspace, every pre-image is too, and taking a basis or adding elements.	Only $T^{-1}(\vec{0})$ is a subspace. Represent a general pre-image as an offset plus a subspace and confine subspace operations to the offset-free part.
Reading $T^{-1}$ as an inverse function	high	The notation suggests invertibility, so a set-valued object is treated as a single vector.	Check injectivity and surjectivity first. The pre-image notation is defined for every linear transformation; the inverse function exists only for bijections.
Reporting a single command for a positive-dimensional pre-image	medium	A solver returns one representative and it is recorded as if it were unique.	Always report the nullity alongside the solution and state the criterion used to select the representative, since different libraries return different members.
Missing an inconsistency because only the coefficient matrix was reduced	high	Row-reducing $A$ without the target column discards the information that detects an unreachable target.	Always reduce the augmented array and test whether the final column is a pivot column before interpreting anything else.
Exact-zero rank test on a measured operator	high	A near-dependent column is classified as independent, so a genuinely redundant system is reported as having a unique solution with an enormous command magnitude.	Determine rank from singular values against a tolerance derived from the measurement uncertainty, and use a truncated decomposition to obtain the representative.
Assuming the minimum-norm command is operationally best	medium	The pseudoinverse default is adopted without checking actuator limits, wear or differing authority.	Formulate the selection as an explicit optimisation over the coset with the relevant weights and constraints, and record the criterion in the design documentation.
Ignoring the null space in fault diagnosis	medium	A fault whose signature lies in the kernel produces no change in any monitored output and is therefore undetectable.	Compute the kernel of the sensing map during design and confirm that no credible fault mode lies within it; add sensing channels if one does.
Recomputing the kernel for every new target	low	Treating each pre-image as an independent problem when only the target has changed.	The kernel depends on the operator alone. Compute it once, retain the factorisation, and recompute only the particular element per target.

FAQs

Does the notation

T^{-1}(\vec{v})

mean the transformation is invertible?

No. It is set notation for the collection of all inputs mapped to $\vec{v}$ , and it is defined for every linear transformation, including ones that are far from invertible. Only when $T$ is bijective does every pre-image contain exactly one element, at which point $T^{-1}$ can also be read as a function.

Why is a pre-image not a subspace?

Because a subspace must contain the zero vector, and $T(\vec{0}) = \vec{0}$ , so $\vec{0}$ lies in $T^{-1}(\vec{v})$ only when $\vec{v} = \vec{0}$ . It also fails closure: adding two elements of $T^{-1}(\vec{v})$ gives a vector mapping to $2\vec{v}$ . The pre-image of the zero vector, the kernel, is the sole exception and is a genuine subspace.

How do I know whether a pre-image is empty?

The pre-image of $\vec{v}$ is empty exactly when $\vec{v}$ lies outside the range of $T$ . For a matrix representation, row-reduce the augmented array $[\, A \mid \vec{v} \,]$ and check the final column: a leading one there means the system is inconsistent and the pre-image is empty.

Why do all non-empty pre-images have the same dimension?

Because each is a translate of the same subspace, the kernel. If $\vec{u}_1$ and $\vec{u}_2$ both map to $\vec{v}$ , their difference maps to $\vec{0}$ , so the whole pre-image is one particular element plus the kernel. Translating a subspace does not change its dimension, so every non-empty pre-image has dimension equal to the nullity.

What is the relationship to the particular-plus-homogeneous solution of a linear system?

They are the same statement. The solution set of $A\vec{x} = \vec{b}$ is the pre-image of $\vec{b}$ under the transformation $\vec{x} \mapsto A\vec{x}$ , and the particular solution plus null space description is exactly the coset structure. Working with pre-images generalises the result from matrices to transformations between arbitrary vector spaces.

If a pre-image contains many commands, which one should I use?

Every element achieves the target equally well, so a secondary criterion decides. The minimum Euclidean norm element is the default returned by a pseudoinverse and is orthogonal to the kernel, but it ignores actuator limits and unequal costs. Parametrise the coset by the kernel basis and optimise the criterion that actually matters.

How do pre-images relate to injectivity and surjectivity?

A transformation is injective when no pre-image has more than one element, which is equivalent to a trivial kernel. It is surjective when no pre-image is empty, equivalent to the range filling the codomain. It is bijective when every pre-image is a single point, which is exactly when the inverse function exists.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section LT, subsection PI. Licensed under the GNU Free Documentation License v1.2.
Axler, S. Linear Algebra Done Right, 3rd edition. Springer, 2015. Chapter 3 on null spaces, ranges and quotients.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013. Chapter 5 on least squares and minimum-norm solutions.
Ben-Israel, A. and Greville, T. N. E. Generalized Inverses: Theory and Applications, 2nd edition. Springer, 2003.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.

AI Suggested Questions

Show me a linear transformation with a two-dimensional kernel and describe the pre-image of a specific reachable target as an explicit plane in the domain.
Derive the minimum-norm element of a pre-image from the pseudoinverse and show it is orthogonal to the kernel.
How does a redundancy resolution scheme in robotics choose a point in the pre-image of a desired end-effector velocity, and what happens near a singular configuration?
Explain why the pre-images of a linear transformation form the equivalence classes of a relation, and connect that to the rank-nullity theorem.
Given a measured operator with one very small singular value, compare the pre-image computed by exact solve against the one from a truncated singular value decomposition.
Work through syndrome decoding for a small linear code and identify the pre-image and the coset representative explicitly.

Related Calculators

Pre-Image Calculator

Row-reduce $[\, A \mid \vec{v} \,]$ to report emptiness, a particular element and a kernel basis in a single pass.

Null Space Basis Calculator

Produce a basis for the kernel and its dimension, with a selectable rank tolerance for measured operators.

Minimum-Norm Solution Calculator

Return the minimum Euclidean norm element of a pre-image, with optional weighting for unequal actuator costs.

Overview

Definition

Pre-Image

Kernel

Range

Coset of a Subspace

Concepts

Emptiness is a reachability question

A non-empty pre-image is a translate of the kernel

A pre-image is almost never a subspace

Pre-images partition the domain

Pre-image sizes characterise injectivity and surjectivity

Computing a pre-image in practice

Choosing a representative from the pre-image

Procedure: determine a pre-image

Equations

Variable Definitions

Worked Numerical Example

Problem statement

State the transformation

Set up the pre-image of the target pattern

Read the consistency verdict and the structure

Extract a particular command

Extract the kernel and assemble the pre-image

Test a second, unreachable target

Choose the minimum-effort command

Report the selected command

Applications &amp; Industry Use

Redundancy resolution and null-space motion

Distributing a command among redundant actuators

Feasible feed combinations for a target product

Non-uniqueness in reconstruction

Self-stress states in indeterminate structures

Syndrome decoding

Design Considerations

Answer the emptiness question before anything else

Do not treat a pre-image as a subspace

Choose the representative deliberately

Nullity is the size of the design freedom

Rank is a tolerance decision on measured operators

Optimise over the coset, not over the whole domain

Standards &amp; Codes

Material Selection

Manufacturing Notes

One reduction answers everything

Doing it by hand

Library behaviour and the minimum-norm default

Verification that catches real errors

Cost of selecting a representative

Failure Modes &amp; Common Mistakes

FAQs

Related Engineering Topics

References

AI Suggested Questions

Related Calculators

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Applications & Industry Use

Standards & Codes

Failure Modes & Common Mistakes