Matrices

Bookkeeping for linear combinations#

Arrange vectors as columns to get the matrix . A linear combination then becomes a matrix–vector product:

This is just another way to write a linear combination.

Matrix multiplication#

Matrix multiplication

Let be an matrix and an matrix. The product is the matrix with entries

For example,

Order matters

In general . Even when is defined, may not be defined at all.

Associativity#

Associativity

Whenever the products are defined, .

Proof

Compare the entries of both sides:

The middle step only swaps the order of two finite sums.

Common matrices#