Vector Spaces

If you aren’t feeling totally comfortable with fields, don’t fret. While we can talk about vector spaces over any field, you can think about them as over \mathbb R if that helps you geometrically.

We started out pointing out the importance of \mathbb R and other well-known fields. But if the real line \mathbb R is important, then the real plane \mathbb R^2 should be twice as important! And \mathbb R^3 is important too. That’s “3-dimensional” space, where we live. I use quotation marks, because we don’t know what dimension is yet.

These objects aren’t fields in any obvious way. (How do I multiply points in \mathbb R^3, or take their inverses?). These are what we call vector spaces. A vector space over a field k is closed under some commutative operation addition, and we can multiply by elements from our field. What the hell? This feels so arbitrary. Let’s look at \mathbb R^3 for an example.

If we have two points (x,y,z) and (x',y',z') in \mathbb R^3, we can add them by adding their components. That is

(x,y,z)+(x',y',z')=(x+x',y+y',z+z').

Well, we can also multiply them by elements of our field (in this case, we will use \mathbb R). If r\in\mathbb R, we can scale (x,y,z) by r, by saying that

r\cdot(x,y,z)=(rx,ry,rz).

You should think of the elements of our vector space, called vectors, as pointing in some direction, as the image shows. When we multiply by element in our field, we scale the vectors.

A representation of vectors in R^2

k is sometimes called the scalar field. In the image, the dotted vector B is scaled by the scalar -1 to get the vector labeled -B.

Notice that the “starting point” of a vector isn’t really important. What is important is the displacement it represents.

Let’s write out the rules for a vector space once in full generality. If you ever feel lost, just think about \mathbb R^2 and this picture. Pretty much everything works analogously (with the proper analogy).

A vector space V over a field k has addition and scalar multiplication such that

  • V satisfies all the rules about + for a field (commutativity, associativity, identity, inverses)
  • If  s\in k and v\in V, then s\cdot v\in V.
  • If u,v\in V, and s\in k, s\cdot (u+v)=s\cdot u+s\cdot v.
So right now you are in one of two categories. If you totally get this, congrats. If you don’t, you are not alone. The easiest thing to do is think about \mathbb R^2, and verify that this is a vector space with the scalar field \mathbb R. When I work out examples, I’ll probably do so in \mathbb R^2 or \mathbb R^3.
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16 Responses to Vector Spaces

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