Linear Algebra 14 | Column Picture of the Matrix-Vector Product [dark version]

TL;DR

An M×N matrix A can be treated as N column vectors A1, A2, …, AN, each in RM.

Briefing Cornell Notes

Briefing

A matrix can be understood as a “column machine” that turns an input vector into an output vector by forming a linear combination of its columns. Instead of treating the matrix as just a table of numbers, the column picture renames each column of an M×N matrix A as A1, A2, …, AN, where each Ai is a column vector with M components. This reframing makes the matrix–vector product feel less like a mechanical multiplication and more like a structured recipe: the output is built from the columns themselves.

With that setup, the matrix–vector product A·x becomes a sum of scaled columns. For a vector x in RN with components (x1, x2, …, xN), the product A x can be written as x1·A1 + x2·A2 + … + xN·AN. The key detail is dimensional: each Ai lives in RM, so every scaled column also lives in RM, and their sum stays in RM. That’s why the result A x always lands in RM when A is M×N and x has N components.

This column-combination viewpoint also clarifies what a matrix does conceptually. The matrix acts like a transformation (a “box”) that takes an input vector x from RN and outputs a vector in RM. In mathematical terms, the matrix defines a map fA: RN → RM by the rule fA(x) = A x. The important takeaway is that the information stored in the matrix is exactly the information stored in the map: the same data that appear as entries in the table of A determine how every input vector is transformed.

Finally, the column picture sets up a broader theme for the course: matrices and linear maps are two ways of describing the same underlying transformation. The column picture makes it explicit that A x is always a linear combination of the columns of A, with the scalars given by the components of x. Later work will confirm that this map is linear, but the structure is already visible in the way the output is assembled from scaled columns.

The discussion ends by previewing a complementary perspective—the row picture—promising that the next step will show an analogous interpretation using rows instead of columns.

Cornell Notes

An M×N matrix A can be viewed as a collection of N column vectors A1, …, AN, each with M components. For an input vector x = (x1, …, xN) in RN, the product A x equals the linear combination x1·A1 + x2·A2 + … + xN·AN. Because each Ai lies in RM, the sum always produces an output in RM. This makes the matrix–vector product a transformation from RN to RM, captured by the map fA: RN → RM defined by fA(x) = A x. The matrix’s numerical entries and the map’s action contain the same information, and the output’s dependence on x is through scaling the columns by x’s components.

How does the column picture rename the structure of an M×N matrix A?

Instead of seeing A as a single table, treat it as N separate columns. Label the first column A1, the second A2, and so on up to AN. Each column vector Ai has M components, so Ai ∈ RM. This turns the matrix into a “set of column vectors” that can be individually scaled and added.

Why can A x be written as a sum of scaled columns, and what are the scalars?

Let x ∈ RN with components (x1, …, xN). The matrix–vector product can be expressed as x1·A1 + x2·A2 + … + xN·AN. The scalars in this linear combination are exactly the components of x: x1 scales A1, x2 scales A2, and so on through xN scaling AN.

What dimensional check guarantees that A x lands in RM?

Each column Ai is an M-component vector, so Ai ∈ RM. Scaling a column by a scalar keeps it in RM, and adding vectors in RM stays in RM. Since A x is a sum of scaled columns Ai, the result must lie in RM.

How does the column picture connect matrices to linear maps?

The transformation viewpoint defines a map fA: RN → RM by fA(x) = A x. The column picture shows that for every input x, the output is assembled as a linear combination of A’s columns, with coefficients taken from x. That correspondence is what makes the matrix and the map carry the same information about the transformation.

What does it mean to say A x is always a linear combination of the columns of A?

For any input vector x, the output A x is constructed only by scaling the columns of A and summing them. There’s no other mechanism involved: the coefficients come directly from x’s components, so the dependence on x is through linear scaling of the fixed column vectors.

Review Questions

Given A is M×N and x has N components, write the column-picture formula for A x using A1, …, AN.
If each column Ai lies in RM, what can be concluded about the space containing A x?
How does the map fA: RN → RM relate to the matrix–vector product in the column picture?

Key Points

1
An M×N matrix A can be treated as N column vectors A1, A2, …, AN, each in RM.
2
For x = (x1, …, xN) ∈ RN, the product A x equals x1·A1 + x2·A2 + … + xN·AN.
3
The coefficients in the linear combination are precisely the components of x.
4
Because each Ai ∈ RM, the output A x always lies in RM.
5
A matrix defines a transformation (map) fA: RN → RM via fA(x) = A x.
6
The matrix’s numerical entries and the map’s action encode the same transformation information.
7
The column picture sets up a parallel “row picture” perspective for later comparison.

Highlights

A x is not just a computation—it is a linear combination of A’s columns, with x’s entries as the weights.

Dimensional consistency is built in: M×N times N×1 always produces an M-dimensional output.

The matrix–vector product can be reframed as a map fA: RN → RM, making the transformation viewpoint explicit.

Renaming columns as A1, …, AN turns the product into a clear “scale-and-add” recipe. 

Topics

Column Picture
Matrix-Vector Product
Linear Combination
Linear Maps
Dimensionality