Linear Algebra 34 | Range and Kernel of a Matrix

TL;DR

Range(A) is the set of all vectors of the form Ax and always lies in R^m.

Briefing Cornell Notes

Briefing

Range and kernel are the two core subspaces that determine whether a linear system has solutions and whether those solutions are unique. For an m×n matrix A, the associated linear map f_A sends vectors in R^n to vectors in R^m. The range of A is the set of all outputs the map can produce—formally, Range(A) = {Ax : x ∈ R^n}—so it lives inside R^m. This “reachable outputs” viewpoint matters because it directly controls solvability: in a system Ax = b, a solution can exist only if b lies in Range(A).

The kernel of A answers a different question: which inputs get sent to the zero vector. The kernel is a subset of R^n defined by Ker(A) = {x ∈ R^n : Ax = 0}. It can also be described as the preimage of the singleton {0} under the map f_A, meaning it collects all vectors that collapse to zero after multiplication by A. Like the range, the kernel is always a subspace, so it always contains the zero vector; in extreme cases it can be as small as {0} or as large as all of R^n.

A key structural fact links these abstract definitions to the geometry of the matrix itself. The range of A can be expressed as the span of A’s column vectors. If A has columns A1, A2, …, A_n, then Range(A) equals span{A1, A2, …, A_n}. This matches the column picture of matrix-vector multiplication: multiplying A by x produces a linear combination of the columns, so every vector in the range is built from those columns, and every such linear combination is achievable by choosing an appropriate x.

Together, range and kernel form the backbone of the solution theory for linear equations. For Ax = b, Range(A) determines existence: if b is not in the range, no x can satisfy the equation. Kernel(A) determines uniqueness: if the kernel contains only the zero vector (often called “trivial”), then solutions—when they exist—are unique. If the kernel is bigger than {0}, then multiple inputs map to the same output, which leads to non-uniqueness (and, in typical cases, infinitely many solutions). The lesson ties these notions to the practical “calculation machine” that will come next, where explicit examples will show how range and kernel behave for concrete matrices.

Cornell Notes

For an m×n matrix A, the associated linear map f_A : R^n → R^m produces two fundamental subspaces. The range (Range(A)) is the set of all outputs Ax can generate, so Range(A) = {Ax : x ∈ R^n} ⊆ R^m. The kernel (Ker(A)) is the set of all inputs that produce the zero vector, so Ker(A) = {x ∈ R^n : Ax = 0} ⊆ R^n. Both are always subspaces, meaning they include the zero vector. Range controls existence of solutions to Ax = b (b must lie in Range(A)), while kernel controls uniqueness (a trivial kernel implies uniqueness; a larger kernel implies non-uniqueness).

How is the range of a matrix A defined, and where does it live?

Range(A) is defined as the set of all vectors you can reach as outputs of the linear map f_A. Concretely, Range(A) = {Ax : x ∈ R^n}. Because A has m rows, every Ax lies in R^m, so Range(A) ⊆ R^m.

What is the kernel of a matrix A, and what does it mean in terms of the map f_A?

Ker(A) is the set of all vectors x that get sent to the zero vector: Ker(A) = {x ∈ R^n : Ax = 0}. In map language, it is the preimage of {0} under f_A, so it collects every input that collapses to the same output, namely 0.

Why is Range(A) equal to the span of A’s columns?

Let A have columns A1, A2, …, A_n. Multiplying A by a vector x produces a linear combination of these columns. Therefore every vector in Range(A) lies in span{A1, …, A_n}. Conversely, any linear combination of the columns can be realized as Ax for a suitable choice of x, so Range(A) equals that span.

How does the range determine whether the linear system Ax = b has a solution?

A solution exists only if b is an achievable output of the map f_A. Since outputs are exactly the vectors in Range(A), the system Ax = b is solvable only when b ∈ Range(A). If b is outside the range, no x can satisfy the equation.

How does the kernel relate to uniqueness of solutions for Ax = b?

If two different solutions x1 and x2 both satisfy Ax = b, then A(x1 − x2) = 0, meaning x1 − x2 lies in Ker(A). So uniqueness corresponds to having no nonzero vectors in the kernel. A trivial kernel (Ker(A) = {0}) implies uniqueness; a larger kernel implies multiple solutions (typically infinitely many when solutions exist).

Review Questions

For a given matrix A, what condition on b guarantees that Ax = b has at least one solution?
State the definitions of Range(A) and Ker(A) and specify which ambient spaces they belong to.
Explain, using the column-span idea, why every vector in Range(A) can be written as a linear combination of A’s columns.

Key Points

1
Range(A) is the set of all vectors of the form Ax and always lies in R^m.
2
Ker(A) is the set of all vectors x such that Ax = 0 and always lies in R^n.
3
Both Range(A) and Ker(A) are subspaces, so they always contain the zero vector.
4
Range(A) equals the span of A’s column vectors, linking the map’s outputs to the matrix’s columns.
5
In Ax = b, solvability requires b ∈ Range(A); otherwise no solution exists.
6
Uniqueness of solutions is tied to the kernel: a trivial kernel supports uniqueness, while a nontrivial kernel leads to non-uniqueness (often infinitely many solutions).

Highlights

Range(A) = {Ax : x ∈ R^n} captures every output a matrix can produce, and that directly governs whether Ax = b can be solved.

Ker(A) = {x ∈ R^n : Ax = 0} collects all inputs that collapse to zero, which is the mechanism behind non-unique solutions.

Range(A) is exactly the span of the columns of A, turning an abstract definition into a concrete construction.

Both range and kernel are always subspaces, so their structure is guaranteed rather than optional.

Topics

Range of a Matrix
Kernel of a Matrix
Linear Maps
Solving Linear Systems
Column Span