Kernel & Range — Kernel and Range (Image) of a Linear Transformation - Linear Algebra

SECTION 1

Historical Context & Motivation

When mathematicians first studied systems of equations, they noticed something curious. Some systems had many solutions, while others had none at all. Understanding why this happened led to two powerful ideas: the kernel (what gets squished to zero) and the range (what outputs are possible). These ideas are central to linear algebra, a branch of math that powers everything from video game graphics to machine learning.

1750s • Early Systems of Equations

Gabriel Cramer and others studied when systems of linear equations have solutions. They noticed patterns connected to the coefficients.

1844 • Grassmann's Linear Extension Theory

Hermann Grassmann published ideas about vector spaces and transformations, laying the groundwork for the kernel and range concepts.

1888 • Peano Defines Vector Spaces

Giuseppe Peano formally defined vector spaces, giving mathematicians the language to describe kernels and ranges precisely.

1900s–1940s • Modern Linear Algebra Takes Shape

Mathematicians unified the ideas of kernel and range into the Rank–Nullity Theorem, one of the most important results in linear algebra.

Here is the key question these mathematicians were trying to answer: given a linear transformation (a special kind of function that uses multiplication and addition to map inputs to outputs), which inputs get mapped to the zero vector, and which outputs can the function actually reach? Understanding these two questions unlocks a deep understanding of how transformations behave.

SECTION 2

Core Principles & Definitions

Before we dive in, let's make sure we're on the same page about a few ideas. A linear transformation is a function T that takes vectors from one space (the domain) and maps them to vectors in another space (the codomain). It follows two rules: T(u + v) = T(u) + T(v) and T(c·v) = c·T(v). Think of it as a machine that stretches, rotates, or flattens vectors — but never bends or breaks them.

Kernel (Null Space)

The set of all input vectors that the transformation sends to the zero vector. Written as ker(T) = { v : T(v) = 0 }.

Range (Image)

The set of all possible output vectors the transformation can produce. Written as range(T) = { T(v) : v is in the domain }.

Nullity

The dimension (number of independent directions) of the kernel. A bigger nullity means more inputs collapse to zero.

Rank

The dimension of the range. It tells you how many independent output directions the transformation actually uses.

Rank–Nullity Theorem

The rank plus the nullity always equals the dimension of the domain: rank + nullity = dim(domain). Nothing is lost or gained — it all adds up.

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KEY TAKEAWAY

Imagine a projector casting a 3D object's shadow onto a flat wall. The kernel is like the direction the projector light travels — anything pointing purely in that direction casts no shadow (maps to zero). The range is the set of all shadows you can see on the wall — the actual outputs of the projection.

SECTION 3

Visual Explanation

The diagram below shows a linear transformation T mapping vectors from a 2D input space (the domain, on the left) to a 2D output space (the codomain, on the right). Notice how certain input vectors all land on the zero vector in the output — those inputs form the kernel. The colored region on the right shows the range — the set of outputs the transformation can actually reach.

The cyan dashed line in the domain represents the kernel — all four vectors v₁ through v₄ get mapped to the zero vector. The pink line in the codomain represents the range — the set of all reachable outputs.

Look at the left ellipse (the domain). The cyan dots along the dashed line represent vectors that all get sent to the single zero vector on the right. That whole collection is the kernel. Meanwhile, vectors like u₁ and u₂ (pink and purple dots) get sent to specific nonzero outputs on the right. The line those outputs lie on is the range. If the transformation "flattens" a 2D space into a 1D line, the range is that line and the kernel is the direction that got squished flat.

SECTION 4

Mathematical Framework

Most linear transformations can be represented by a matrix (a grid of numbers). If we call the matrix A, then the transformation T(v) = A·v multiplies the matrix by a vector. Finding the kernel and range becomes a problem of solving equations with that matrix.

KERNEL DEFINITION

ker(T) = { v ∈ ℝⁿ : A·v = 0 }

The kernel is the solution set of the homogeneous equation A·v = 0. You find it by row-reducing the matrix A and solving for the free variables.

RANGE DEFINITION

range(T) = { A·v : v ∈ ℝⁿ } = column space of A

The range equals the column space of A — the span of all column vectors in the matrix. Every output is a combination of those columns.

RANK–NULLITY THEOREM

rank(A) + nullity(A) = n

Here n is the number of columns of A (the dimension of the domain). Rank is the dimension of the range, and nullity is the dimension of the kernel. They always add up to n.

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How to Find the Kernel & Range

Kernel: Write the augmented matrix [A | 0] and row-reduce to reduced row echelon form (RREF). The free variables give you the kernel vectors. Range: Row-reduce A. The columns that contain pivots (leading 1s) in RREF tell you which original columns of A form a basis for the range.

SECTION 5

How Kernel & Range Relate — A Detailed Breakdown

The kernel and range are like two sides of the same coin. If the kernel is large, the transformation crushes many inputs to zero and the range is small. If the kernel is tiny (just the zero vector itself), the transformation preserves more information and the range is as large as possible. The Rank–Nullity Theorem captures this tradeoff perfectly.

Three transformations from ℝ³ illustrate the Rank–Nullity Theorem. As the kernel grows, the range shrinks — and their dimensions always add up to 3.

Property	Kernel (Null Space)	Range (Image / Column Space)
What it answers	Which inputs get mapped to 0?	Which outputs can T produce?
How to find it	Solve A·v = 0 by row-reducing	Identify pivot columns after row-reducing A
Its dimension is called	Nullity	Rank
If it's large, then…	Many inputs collapse; T loses info	T covers many output directions
Always a subspace?	Yes, of the domain	Yes, of the codomain

Side-by-side comparison of kernel and range

SECTION 6

Worked Example

Let's find the kernel and range of the linear transformation T: ℝ³ → ℝ² defined by the matrix A below. We'll row-reduce the matrix, identify the kernel, and then determine the range.

GIVEN MATRIX

A = [ 1 2 1 ; 2 4 2 ]

This is a 2 × 3 matrix, so T maps ℝ³ to ℝ². Row 1 is [1, 2, 1] and Row 2 is [2, 4, 2].

Finding the Kernel and Range of A

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Step 1 — Set Up the Homogeneous System

To find the kernel, we solve A·v = 0. Write the augmented matrix [A | 0]: [ 1 2 1 | 0 ] [ 2 4 2 | 0 ]
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Step 2 — Row-Reduce to RREF

Replace Row 2 with Row 2 − 2 × Row 1: [ 1 2 1 | 0 ] [ 0 0 0 | 0 ] Row 2 is now all zeros. The matrix is in reduced row echelon form with one pivot in column 1.

One pivot column (column 1). Two free variables: x₂ and x₃.
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Step 3 — Solve for the Kernel

From Row 1: x₁ + 2x₂ + x₃ = 0, so x₁ = −2x₂ − x₃. Let x₂ = s and x₃ = t (free parameters). Then the general solution is: v = s·(−2, 1, 0) + t·(−1, 0, 1)

ker(T) = span{(−2, 1, 0), (−1, 0, 1)}. The kernel is a 2-dimensional plane in ℝ³. The nullity = 2.
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Step 4 — Find the Range

The range equals the column space of A. Since column 1 has a pivot, the original column 1 of A is a basis vector for the range. The original column 1 is (1, 2). Columns 2 and 3 have no pivots, so they are dependent on column 1.

range(T) = span{(1, 2)}. The range is a line in ℝ². The rank = 1.
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Step 5 — Verify with Rank–Nullity

The domain is ℝ³, so n = 3. Check: rank + nullity = 1 + 2 = 3 ✓. Everything checks out.

rank(1) + nullity(2) = 3 = number of columns ✓

SECTION 7

Important Properties & Common Pitfalls

Property / Fact	What It Means
Kernel always contains the zero vector	T(0) = 0 for any linear transformation, so 0 is always in the kernel.
T is one-to-one ⟺ kernel = {0}	If only the zero vector maps to zero, then no two different inputs produce the same output.
T is onto ⟺ range = entire codomain	Every vector in the codomain is an output of some input.
Kernel and range are both subspaces	They're closed under addition and scalar multiplication — they're not just random collections.
Rank–Nullity always holds	You can't increase the range without shrinking the kernel, and vice versa.

Key properties of the kernel and range

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COMMON PITFALL

Students sometimes confuse the kernel (a subset of the domain) with the range (a subset of the codomain). Remember: the kernel lives where the inputs live, and the range lives where the outputs live. Think of it like a mail sorting machine: the kernel is the pile of letters that get shredded (sent to 'zero'), and the range is the set of mailboxes that actually receive something.

SECTION 8

Connections to Advanced Topics

The kernel and range are your entry point into deeper ideas in linear algebra and beyond. Once you understand these, you can tackle eigenvalues, invertibility, and even applications in data science. Here's how the concepts connect.

This Lesson's Concept	Advanced Extension
Kernel = {0} means T is one-to-one	An invertible matrix has a trivial kernel — you can "undo" the transformation.
Rank–Nullity Theorem	Generalizes to the First Isomorphism Theorem in abstract algebra: domain/kernel ≅ range.
Dimension of the kernel	Tells you how many free variables exist in a system — central to understanding solution sets.
Column space (range)	In data science, the column space reveals the directions of maximum variance (PCA).

How kernel and range connect to more advanced topics

As you continue studying linear algebra, you'll see that virtually every major result — from the determinant to eigenvalues — connects back to how a transformation's kernel and range behave. Mastering them now gives you a powerful lens for everything that comes next.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

In your own words, explain the difference between the kernel and the range of a linear transformation. Where does each one "live" — in the domain or the codomain?

Show answer

The kernel is the set of all input vectors that the transformation maps to the zero vector; it lives in the domain. The range is the set of all possible output vectors that the transformation can produce; it lives in the codomain. Think of the kernel as the inputs that "disappear" and the range as the outputs that "show up." Both are subspaces — they're closed under addition and scalar multiplication.

Question 2: BASIC CALCULATION

Let A = [ 1 3 ; 0 0 ]. Find the kernel of the linear transformation T(v) = A·v, where v is in ℝ².

Show answer

We solve A·v = 0. The system gives us x₁ + 3x₂ = 0 (from row 1) and 0 = 0 (from row 2). So x₁ = −3x₂. Let x₂ = t, then v = t·(−3, 1). Therefore ker(T) = span{(−3, 1)}, which is a line through the origin in ℝ². The nullity is 1, and the rank is 1, so rank + nullity = 1 + 1 = 2, which equals the number of columns.

Question 3: INTERMEDIATE

Consider the matrix A = [ 1 0 2 ; 0 1 −1 ; 0 0 0 ]. Find both the kernel and the range of T(v) = A·v. Then verify the Rank–Nullity Theorem.

Show answer

Row-reducing A (it's already in RREF), we see pivots in columns 1 and 2. Column 3 is free. From the system: x₁ = −2x₃ and x₂ = x₃. Let x₃ = t, so ker(T) = span{(−2, 1, 1)}, giving nullity = 1. The range is the column space spanned by the original columns 1 and 2 of A: range(T) = span{(1, 0, 0), (0, 1, 0)}, giving rank = 2. Verification: rank(2) + nullity(1) = 3 = number of columns ✓.

Question 4: APPLIED

A 2D video game engine uses the matrix A = [ 1 0 ; 0 0 ] to project objects onto the x-axis. Find the kernel and range of this transformation. What happens to a character standing at position (3, 5)?

Show answer

Solving A·v = 0: x₁ = 0 and 0·x₂ = 0, so x₂ is free. The kernel is ker(T) = span{(0, 1)} — the entire y-axis. Any vector pointing straight up or down gets squished to zero. The range is span{(1, 0)} — the x-axis. The rank is 1 and the nullity is 1, with rank + nullity = 2 ✓. The character at (3, 5) maps to (3, 0), so they're projected onto the x-axis. The y-coordinate (height) is lost because the y-direction is entirely in the kernel.

Question 5: CRITICAL THINKING

Suppose T: ℝ⁴ → ℝ³ is a linear transformation. Can the kernel of T be just the zero vector? Use the Rank–Nullity Theorem to justify your answer.

Show answer

If the kernel were {0}, the nullity would be 0. By the Rank–Nullity Theorem, rank + nullity = 4, so the rank would need to be 4. But the rank is the dimension of the range, which is a subspace of ℝ³. A subspace of ℝ³ can have dimension at most 3. Since 4 > 3, the rank cannot be 4. Therefore the kernel cannot be just {0} — the nullity must be at least 1. This means any linear map from a higher-dimensional space to a lower-dimensional space must "collapse" at least some nonzero inputs to zero.

SUMMARY

Lesson Summary

Summary

A linear transformation T maps input vectors to output vectors using matrix multiplication. The kernel (or null space) is the set of all inputs that T sends to the zero vector — you find it by solving A·v = 0. Its dimension is called the nullity. The range (or image) is the set of all possible outputs — it equals the column space of the matrix, and its dimension is called the rank.

The Rank–Nullity Theorem ties everything together: rank + nullity = number of columns in A. A larger kernel means more information is lost and the range shrinks. If the kernel is trivial (only contains the zero vector), the transformation is one-to-one and no information is lost. These ideas are the foundation for understanding invertibility, eigenvalues, and many real-world applications from data science to engineering.