This page is a preview of the main words you will study in Chs 3-6. I'd like you to start getting used to them now. Later, we will learn precise definitions of them and how to calculate with them.

Take a look at the four arrows, or vectors, x, y, z and v. The 3D space they belong to is called a vector space, which means the vectors can be added together (etc). For example, adding x to z makes a new vector with the same direction as v, that's just a bit longer. See it?

Notice that x is perpendicular, or orthogonal, to z. Both are orthogonal to y. So, this set of three vectors is mutually orthogonal. The set of all four vectors is not mutually orthogonal because v is not orthogonal to x (or to z).

I created v by rotating x about 45 degrees. This rotation is an example of a linear transformation, which means (in part) that it is a function L, from vectors to vectors. And L(x) = v. [Analogy: You can create 6 from 4 by adding 2. "Adding 2" is a function f(x) = x+2, and f(4)=6. By the way, there is a formula for L, discussed in Ch 4]. What if you apply L to v? Answer: L(v) = z.

What if you apply L to y? Since y is on the axis of rotation, it would not move at all ! L(y) = y. So, y is an example of an eigenvector of L [which just means L(y) is parallel to y]. Basically, this means y is an important vector for understanding L.

We'll learn in Ch 4 to calculate a matrix to go with L. In Ch 5, we'll learn about orthogonality. And in Ch 6, we'll learn how to find L's eigenvectors.

Notice that x, v and z lie in the same 2D plane (this plane is a subset of the 3D vector space, and is an example of a subspace). In fact, v is a linear combination of the other two, because we could shorten x and z a little, and then add them, to make v. By shortening (or lengthening or reversing) x and z in various ways before adding, we could actually make any vector in the 2D plane. So, we say x and z span this plane.

It is also true that the three vectors v, x and z span the plane. But since v is a linear combination of the others, we don't really need it, and we say that this set of three vectors is linearly dependent. But x and z don't have this problem, so they are linearly independent and together they form a basis for the plane. They aren't the only possible basis (for example, x and v form a basis too, but mutually orthogonal vectors are usually cleaner to work with).

A standard basis for the 3D space would be x, y and z (because they span 3D space and are linearly independent and are mutually orthogonal). An inferior basis such as x, y and v can always be "fixed" (for example, by rotating v around until it becomes z). The repair is usually done by the Gram Schmidt Process.

Please read this over a few times, and try to retain some of the vocabulary. But do not assume that you know anything yet! I haven't scratched the surface. Learning the precise definitions and using them correctly will be quite important.

Back to the Help page