Rank-Nullity Theorem

Core idea

Overview

In the context of a linear map T: V → W where V is finite-dimensional, this theorem provides a fundamental constraint on the relationship between the dimensions of the kernel and the image.

When to use: This theorem is the most fundamental tool in undergraduate linear algebra for determining the dimensions of subspaces associated with linear transformations.

Why it matters: It links the concept of injectivity (connected to the nullity) and surjectivity (connected to the rank) to the geometry of the domain space.

Symbols

Variables

$dim$ (V) = Dimension of Domain, $rank$ (T) = Rank, $nullity$ (T) = Nullity

dim (V)

Dimension of Domain

Variable

rank (T)

Rank

Variable

nullity (T)

Nullity

Variable

Walkthrough

Derivation

Derivation/Understanding of Rank-Nullity Theorem

This derivation shows that for a linear transformation, the sum of the dimension of its kernel (nullity) and the dimension of its image (rank) equals the dimension of its domain.

V and W are vector spaces over the same field F.
T: V $\to$ W is a linear transformation.
V is a finite-dimensional vector space.

1

Define Kernel and Image Dimensions:

We begin by defining the kernel and image of a linear transformation, which are subspaces of the domain and codomain, respectively. Their dimensions are known as the nullity and rank of the transformation.

Let T : V \to W be a linear transformation. \n The kernel of T is Ker (T) = {v \in V ∣ T (v) = 0_{W}} . \n The image of T is Im (T) = {w \in W ∣ \exists v \in V s.t. T (v) = w} . \n We define nullity (T) = dim (Ker (T)) and rank (T) = dim (Im (T)) .

2

Construct a Basis for the Domain:

We start with a basis for the kernel and extend it to form a complete basis for the entire domain vector space V. This allows us to express any vector in V as a linear combination of these basis vectors.

Let {v_{1}, \dots, v_{k}} be a basis for Ker (T), so k = nullity (T) . \n By the Basis Extension Theorem, we can extend this to a basis for V : \n B_{V} = {v_{1}, \dots, v_{k}, v_{k + 1}, \dots, v_{n}}, where n = dim (V) .

3

Show Images of Extended Basis Form a Basis for the Image:

We examine the images of the basis vectors that were not in the kernel. We prove that these images span the entire image space and are linearly independent, thus forming a basis for the image.

Consider the set of vectors S = {T (v_{k + 1}), \dots, T (v_{n})} . \n We show S is a basis for Im (T) . \n 1. S spans Im (T) : Any w \in Im (T) is T (v) for some v \in V . \n v = c_{1} v_{1} + \dots + c_{k} v_{k} + c_{k + 1} v_{k + 1} + \dots + c_{n} v_{n} . \n T (v) = c_{1} T (v_{1}) + \dots + c_{k} T (v_{k}) + c_{k + 1} T (v_{k + 1}) + \dots + c_{n} T (v_{n}) . \n Since v_{1}, \dots, v_{k} \in Ker (T), T (v_{i}) = 0 for i \leq k . \n So T (v) = c_{k + 1} T (v_{k + 1}) + \dots + c_{n} T (v_{n}), meaning S spans Im (T) . \n 2. S is linearly independent: Assume d_{k + 1} T (v_{k + 1}) + \dots + d_{n} T (v_{n}) = 0. \n T (d_{k + 1} v_{k + 1} + \dots + d_{n} v_{n}) = 0. \n This implies d_{k + 1} v_{k + 1} + \dots + d_{n} v_{n} \in Ker (T) . \n So d_{k + 1} v_{k + 1} + \dots + d_{n} v_{n} = e_{1} v_{1} + \dots + e_{k} v_{k} for some scalars e_{i} . \n e_{1} v_{1} + \dots + e_{k} v_{k} - d_{k + 1} v_{k + 1} - \dots - d_{n} v_{n} = 0. \n Since B_{V} is a basis, all coefficients must be zero, so d_{k + 1} = \dots = d_{n} = 0. \n Thus, S is linearly independent and forms a basis for Im (T) .

4

Conclude the Rank-Nullity Theorem:

By counting the number of vectors in the basis for the image, we establish that the rank is equal to the dimension of the domain minus the nullity. Rearranging this equation yields the Rank-Nullity Theorem.

The number of vectors in the basis S for Im (T) is n - k . \n Therefore, dim (Im (T)) = n - k . \n Substituting the definitions: rank (T) = dim (V) - nullity (T) . \n Rearranging, we get the Rank-Nullity Theorem: rank (T) + nullity (T) = dim (V) .

Result

The number of vectors in the basis S for Im (T) is n - k . \n Therefore, dim (Im (T)) = n - k . \n Substituting the definitions: rank (T) = dim (V) - nullity (T) . \n Rearranging, we get the Rank-Nullity Theorem: rank (T) + nullity (T) = dim (V) .

Source: Linear Algebra Done Right by Sheldon Axler

Free formulas

Rearrangements

Solve for $dim (V)$

Rank-Nullity Theorem: Make $dim$ (V) the subject

x + y

Start from the Rank-Nullity Theorem and express $dim$ (V) in terms of the shorthand variables x (rank) and y (nullity).

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

The graph is a linear plot where the sum of the rank and nullity equals the dimension of the domain. Because the dimension of the domain is constant for a fixed linear map, the relationship between rank and nullity forms a straight line with a negative slope.

Graph type: linear

Why it behaves this way

Intuition

Imagine the total 'size' (dimension) of the input space V being split into two complementary parts by the linear transformation T: one part that gets 'crushed' to the zero vector (the null space), and another part determines the behavior.

rank(T)

The dimension of the image (range) of the linear transformation T. It quantifies the 'output capacity' or the number of independent directions in the output space.

Represents the 'useful' part of the input space that contributes to distinct outputs. A higher rank means the transformation preserves more distinct information.

nullity(T)

The dimension of the kernel (null space) of the linear transformation T. It quantifies the 'information loss' or the number of independent input directions that are mapped corresponding physical or mathematical quantity.

Represents the 'collapsed' part of the input space. A higher nullity means many distinct inputs are mapped to the same output (specifically, zero), indicating significant information loss.

dim(V)

The dimension of the domain vector space V. It represents the total number of independent input components or the 'size' of the input space.

The total 'capacity' of the input information available before the transformation.

Free study cues

Insight

Canonical usage

This equation is used to relate the integer dimensions of vector spaces and linear map properties. The terms 'rank', 'nullity', and 'dimension' refer to the number of basis vectors in the respective spaces, and thus are dimensionless counts.

Common confusion

A common confusion is to associate the mathematical concept of 'dimension' (as in the number of basis vectors) with physical dimensions (e.g., length, mass, time) that carry units.

Dimension note

All quantities in the Rank-Nullity Theorem (rank, nullity, and dimension of V) are mathematical dimensions, meaning they are non-negative integer counts of basis vectors. They do not possess physical units.

Unit systems

rank(T)dimensionless (integer count) - Represents the dimension of the image of the linear transformation T, which is the number of basis vectors in the image space. Must be a non-negative integer.

nullity(T)dimensionless (integer count) - Represents the dimension of the kernel (null space) of the linear transformation T, which is the number of basis vectors in the kernel space. Must be a non-negative integer.

dim(V)dimensionless (integer count) - Represents the dimension of the domain vector space V, which is the number of basis vectors in V. Must be a non-negative integer.

One free problem

Practice Problem

Given a linear transformation T: ℳ → Ⅎ where the kernel is a line through the origin (dimension 1), calculate the rank of T.

Dimension of Domain3

Nullity1

Solve for: $x$

Hint: The dimension of the domain is 3. If the nullity is 1, use the theorem: Rank + Nullity = Dim(V).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In data science, when projecting high-dimensional data into a lower-dimensional space (dimensionality reduction), the Rank-Nullity theorem helps determine the amount of information preserved (rank) versus the information lost (nullity).

Study smarter

Tips

Always verify that the vector space V is finite-dimensional before applying the theorem.
Remember that the dimension on the right side of the equation is the domain, not the codomain.

Avoid these traps

Common Mistakes

Confusing the dimension of the codomain (W) with the dimension of the domain (V).
Assuming the theorem applies to non-linear transformations.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation shows that for a linear transformation, the sum of the dimension of its kernel (nullity) and the dimension of its image (rank) equals the dimension of its domain.

This theorem is the most fundamental tool in undergraduate linear algebra for determining the dimensions of subspaces associated with linear transformations.

It links the concept of injectivity (connected to the nullity) and surjectivity (connected to the rank) to the geometry of the domain space.

Confusing the dimension of the codomain (W) with the dimension of the domain (V). Assuming the theorem applies to non-linear transformations.

In data science, when projecting high-dimensional data into a lower-dimensional space (dimensionality reduction), the Rank-Nullity theorem helps determine the amount of information preserved (rank) versus the information lost (nullity).

Always verify that the vector space V is finite-dimensional before applying the theorem. Remember that the dimension on the right side of the equation is the domain, not the codomain.

References

Sources

Axler, S. (2015). Linear Algebra Done Right.
Axler, Sheldon. Linear Algebra Done Right. Springer, 3rd ed., 2015.
Strang, Gilbert. Introduction to Linear Algebra. Wellesley-Cambridge Press, 5th ed., 2016.
Wikipedia: Rank-nullity theorem
Rank-nullity theorem (Wikipedia article)
Sheldon Axler Linear Algebra Done Right
Gilbert Strang Introduction to Linear Algebra
Wikipedia article 'Rank-nullity theorem'

Overview

Variables

Derivation

Define Kernel and Image Dimensions:

Construct a Basis for the Domain:

Show Images of Extended Basis Form a Basis for the Image:

Conclude the Rank-Nullity Theorem:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Lagrange's Theorem

Orthogonal Projection

Gram-Schmidt Orthogonalization

Frequently Asked Questions

Sources