Ah, vector spaces. The definition reads somewhat like a checklist. “If some elements have these properties, they make up a vector space.” It’s not so hard to verify whether something follows the definition of a vector space — you just check that each property holds. However, it took me some time after the “rules” sunk in, before I could better appreciate the concepts behind them.

The Definition

Informally, vector spaces are defined as a set of elements where you have:

  • Addition
    • Able to add elements together.
    • Closure under addition. (Any two elements added together are also in the set.)
    • Includes an element that gives you the same number when added. (Think 0.)
  • Scalar multiplcation
    • Able to multiply an element by any number. (Take 2 or 5, for example.)
    • Closure under scalar multiplication. (Any element multiplied by a scalar is also in the set.)
    • Includes an element that gives you the same number when multiplied with another element. (Think 1.)
  • A negative value for each positive value
  • The distributive property, where $a(b + c) = ab + ac$

Whew.

Now, would you say the set of real numbers $\reals$ is a vector space?

Hint: Go through the properties above, one by one, and pick out some real numbers to see if that property holds.

For the motivated reader, how about the set of integers? And the set of positive integers?

Stepping Back

I think I missed this the first time, but it’s an important realization. Vector spaces are all about describing the behavior of vectors. (Astute observation, I know.) The definition we saw rightfully captures two key operations vectors are known for. The ability to combine vectors through summation, and to scale them through scalar multiplication.

To answer the question above, yes — the set of real numbers does fall under the definition of a vector space.

But why? Is there a connection between a real number and a vector? Why yes, I’m glad you asked.

It just so happens that a real number can be thought of as a vector of size 1.

For example, you can have $ \begin{bmatrix} 0 \end{bmatrix} $, or $ \begin{bmatrix} 1 \end{bmatrix} $, or $ \begin{bmatrix} \pi \end{bmatrix} $.

It makes sense then, saying that the set of real numbers make up a vector space. Because numbers are vectors, they share vector-like behavior as well.

Conclusion

And that’s all! I hope this post gave a bit more insight on why vector spaces are defined the way they are. Thanks for reading!