4.310 min read

The Column Perspective on Matrix Products

The most geometric way to understand $C = AB$ is column-by-column. Each column of $C$ is the result of multiplying $A$ by the corresponding column of $B$ : $\mathbf{c}_j = A\mathbf{b}_j$ .

So $C = AB$ means: take each column of $B$ , multiply it by $A$ , and the results become the columns of $C$ . This is $n$ separate matrix-vector multiplications.

This perspective makes the composition story concrete: the $j$ -th column of $B$ tells us "where the $j$ -th basis vector goes under $B$ ." Then $A$ transforms that destination. The result is where the $j$ -th basis vector goes under the composition $AB$ .

Formal View

Theorem 4.3 (Column Form)

C = AB

, then the

j

-th column of

C

\mathbf{c}_j = A \mathbf{b}_j

where

\mathbf{b}_j

is the

j

-th column of

B

Interactive Visualization

Matrix Product — Column Perspective

Why This Matters

The column perspective makes it clear that each column of AB is independently computed via matrix-vector products.

Parallelization: each column of $AB$ can be computed independently on separate processors
Sparse matrix products: if $B$ has many zero columns, those columns of $C$ are immediately zero

Learning Resources

Column view of matrix multiplication

MIT OpenCourseWare

Five different ways to view matrix multiplication.

39 min

Quiz

Question 1

The second column of $AB$ equals $A$ times the second column of $B$ .

Common Mistakes

Computing $AB$ by multiplying each row of $A$ with each column of $B$ when the column perspective is clearer.