Linear Transforms, An Introduction


There is a subtle difference between a linear transformation and a linear transform. A transform is always a transformation, but a transformation is not necessarily a transform.

First, a quick review. A linear transformation is a function that maps one vector space into another. It usually maps real space into real space, or complex space into complex space, and is often implemented by a matrix. One space is "transformed" into another, so the function is called a transformation.

When we talk about a transformation we usually mean a linear transformation, but it's best not to assume. Call it a linear transformation, and there will be no confusion. Farther down the page you can call it a transformation, or a function, or a map; but at the top of the page, include the word linear, so there is no doubt.

A function is linear when f(a) + f(b) = f(a+b), and c×f(x) = f(c×x). In other words, f respects addition and scaling. If you want to be fancy, f is a module homomorphism from one R module into another, where R is a division ring. but if you've never heard of a module, or a homomorphism, that's ok.

If you are interested in linear transformations from m dimensional space into n dimensional space, e.g. rotations reflections projections etc, you will find that every possible transformation is implemented by a matrix, and each matrix implements a different function. You will need to understand this function-matrix correspondence before we proceed.

So, what's a transform?

A transform, also called an operator, is a function that operates on functions. Consider the quadratic operator, which I will call q(). It takes a function f and maps it to f squared. Thus q(f) = f2. This is not a linear transform. Consider q(x) + q(x) = 2x2, which is not the same as q(x+x) = 4x2. Transforms don't have to be linear, but when they are, it's not a bad idea to introduce them as "linear transforms" or "linear operators" at the top of the page.

A linear transform, like a linear transformation, acts on a vector space. This time the vector space consists of functions. Since functions can be added and scaled, they form a vector space, and hence they can serve as the input to a linear transform.

Two examples of transforms are differentiation and integration. To keep things simple, let the vector space be the continuous functions on the closed interval [0,1]. I chose a closed interval because there are functions on the open interval, such as 1/x, that are not integrable. If f and g are continuous functions, verify that f+g is continuous, and so is c×f(x). Thus we have a valid vector space. The derivative of f+g is f′+g′, and the derivative of c×f is c×f′. thus differentiation is a linear operator. Show the same for integration.

The domain and range don't have to be the same. Consider the fourier transform, acting on the same vector space. Given a continuous function on [0,1], the result is a sequence of real numbers a0, a1, a2, etc. Well a sequence is a function of sorts, a function on the nonnegative integers, but it doesn't look like a continuous function on [0,1]. Still, the input and output are functions, and we have a valid transform. Furthermore, the sequence produced by f+g is the sum of the sequences for f and for g. And if f is scaled by the constant c, the resulting sequence is scaled by c. Thus a linear transform is born.