Linear Algebra 34 | Range and Kernel of a Matrix [dark version]

TL;DR

Range(A) is the set of all outputs Ax as x varies over R^n, and it is always a subspace of R^m.

Briefing Cornell Notes

Briefing

Range and kernel turn a matrix into two practical “tests” for solving linear systems: whether a right-hand side can be reached at all, and whether the solution is forced to be unique.

Start with an m×n matrix A, which induces a linear map f_A: R^n → R^m via x ↦ Ax. The range of A is the subset of R^m consisting of every vector that can appear as an output of this map. Concretely, Range(A) = {Ax : x ∈ R^n} = A·R^n. In other words, it collects all attainable right-hand sides in the output space. The same idea can be described as the “image” of the map, but the key point is that the range is always a subspace of R^m—so it necessarily contains the zero vector, and it can range from being as small as {0} to being all of R^m.

The kernel of A, written ker(A), lives in the input space R^n. It is the set of all vectors x that get sent to the zero vector: ker(A) = {x ∈ R^n : Ax = 0}. Equivalently, it is the preimage of {0} under f_A. Like the range, the kernel is always a subspace of R^n, again containing the zero vector and potentially spanning everything in extreme cases. Some terminology overlaps with linear-map language: the kernel is also called the null space of A.

A crucial bridge connects these definitions to the geometry of matrices. The range of A can be described using the columns of A. If the columns are A_1, A_2, …, A_n, then Range(A) equals the span of these column vectors. This matches the column picture of matrix-vector multiplication: Ax is always a linear combination of the columns, and every vector in that span can be produced by choosing the right x.

These notions matter immediately for solving linear equations. Consider a linear system A x = b. A solution can exist only if b lies in Range(A); otherwise there is no x that produces b. The kernel, by contrast, governs uniqueness: if ker(A) contains more than just the zero vector, then there is “room” for multiple solutions because different x values can map to the same b. The transcript flags that the exact characterization of solution behavior—no solutions versus infinitely many—will be developed later, but the division of labor is already clear: range controls existence, kernel controls uniqueness.

Cornell Notes

For an m×n matrix A, the induced map f_A: R^n → R^m sends x to Ax. The range of A is Range(A) = {Ax : x ∈ R^n}, a subspace of R^m describing all outputs the system can produce. The kernel is ker(A) = {x ∈ R^n : Ax = 0}, a subspace of R^n describing all inputs that collapse to the zero vector. Range(A) equals the span of A’s columns, linking the abstract definition to the column picture of matrix multiplication. In solving A x = b, b must lie in Range(A) for solutions to exist, while ker(A) determines whether solutions are unique or not.

How is the range of a matrix defined, and what does it mean for solving A x = b?

For A (m×n), Range(A) is the subset of R^m consisting of all vectors of the form Ax as x ranges over R^n. So Range(A) = {Ax : x ∈ R^n}. In the system A x = b, a solution exists only if b is one of those attainable outputs—equivalently, only if b ∈ Range(A). If b is outside the range, no x can produce it.

What is the kernel of a matrix, and how is it related to the zero vector?

The kernel is ker(A) = {x ∈ R^n : Ax = 0}. It collects exactly the inputs that map to the zero vector in R^m. On the level of linear maps, ker(A) is the preimage of {0} under f_A. Because it is a subspace, it always contains the zero vector and can be larger than {0}.

Why does the range equal the span of the columns of A?

Let the columns of A be A_1, A_2, …, A_n. Matrix-vector multiplication expresses Ax as a linear combination of the columns, so every output Ax lies in span{A_1, …, A_n}. Conversely, any vector in that span can be written as a linear combination of the columns, which corresponds to choosing an x that produces it as Ax. Therefore Range(A) = span of the columns.

What does the kernel tell you about uniqueness of solutions?

If ker(A) contains only the zero vector, then there’s no nontrivial way to change x without changing Ax. If ker(A) is larger than {0}, then there exist nonzero vectors v with Av = 0, meaning x and x+v produce the same Ax. That leads to multiple solutions (the transcript notes this will be formalized later).

What are the extreme possibilities for range and kernel?

Both Range(A) ⊆ R^m and ker(A) ⊆ R^n are subspaces. The smallest case for each is {0} (only the zero vector). The largest case is the entire ambient space: Range(A) can be all of R^m, and ker(A) can be all of R^n. The transcript emphasizes that these subspace properties are immediate and always hold.

Review Questions

Given a system A x = b, what condition on b guarantees at least one solution, and which subspace encodes that condition?
If ker(A) contains a nonzero vector v, what does that imply about the relationship between solutions x and x+v?
How can you compute or describe Range(A) using only the columns of A?

Key Points

1
Range(A) is the set of all outputs Ax as x varies over R^n, and it is always a subspace of R^m.
2
ker(A) is the set of all inputs x such that Ax = 0, and it is always a subspace of R^n.
3
Range(A) equals the span of A’s columns, linking the range to the column picture of matrix multiplication.
4
For the system A x = b, solutions can exist only if b lies in Range(A).
5
The kernel determines uniqueness: a nontrivial kernel allows different x values to produce the same b.
6
Both range and kernel are best understood as subspaces arising from the induced linear map f_A: R^n → R^m.

Highlights

Range(A) = {Ax : x ∈ R^n} tells exactly which right-hand sides b are reachable in A x = b.

ker(A) = {x ∈ R^n : Ax = 0} captures all “zero-producing” directions in the input space.

Range(A) is the span of the columns of A, making the abstract definition concrete.

Existence of solutions depends on membership in Range(A), while uniqueness depends on whether ker(A) is trivial. 

Topics

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