Matrix transformations

Earlier, we looked at matrix operators on R2 and R3 – transformations like

and

for example.

We can do the same for Rn: first represent any x in Rn as a column matrix.

A matrix operator on Rn is a transformation of the form x → Ax where A is an n x n matrix.

As before, we can discuss the "geometry" of these operators, decompose them into simpler operators and so on. The basic ideas are the same; the computations just become more complicated.

We can also make the idea of matrix operators more general. Suppose that A is an m x n matrix, not necessarily square. If we continue to represent our vectors by column matrices, then the transformation r → Ar transforms n-tuples into m-tuples – instead of just moving vectors around inside the same space, it transforms them into vectors in a different space.

The way to think about such transformations is to think of them as functions.

Review of functions. Suppose we have two sets A and B.

A function from A to B is a rule f:A → B that assigns to every element a in A a single element f(a) in B.

The set A is called the domain of f, the set B is called the codomain of f, and f is said to map A into B.

The element f(a) in B is called the image of the element a in A, and f is said to map a onto f(a).

Example. Suppose A and B are sets with four elements each.

These rules are functions from A to B.

These rules are not functions from A to B.


This rule maps • into more than one element of B.		This rule does not map • into any element of B.

Multiplication by an m x n matrix transforms each vector in Rn into a single vector in Rm, so it is a rule that defines a function.

A matrix transformation from Rn to Rm is a function T:Rn→ Rm of the form T(r) = Ar for some m x n matrix A. The domain of T is Rn and the codomain of T is Rm.

We can then think of matrix operators as matrix transformations for which the domain and codomain are two copies of the same space Rn.

In the next section, we'll look at these transformations in more detail.