Back to school time again. As I'm discussing all the mathy stuff and insights gained over the summer, I cannot help but notice that many of my peers in second or third year undergrad cannot bridge the gap between vectors and functions.
When I ask them why, the answer I most often get are as follows:
-
Vectors are something learned in linear algebra, and they are
basically pointy arrows in $R^2$ and written with brackets $\{\}$ or sometimes with
arrow on the top -
Functions are completely different, they are the object under study in calculus, never drawn as an arrow, doesn't
satisfy some axioms of vector space, can be drawn in a lot of ways
(piecewise or continuous, with crazy oscillations). -
You can do a lot with functions, such as taking the derivative of it,
or the integral of it. You can find the inverse of a function. -
No authority has ever said functions are vectors, if they did I would
believe them -
Yes, functions are sometimes contained in brackets, but that is just a
vector of functions, not a function
Can someone please provide a good example may bridge this gap? Also, is there any books that explicitly bridge the two concepts?
Thank you for your inputs!
Best Answer
Let me start by giving a unified viewpoint, and then I'll reconcile the other things you heard with it.
Can we consider functions as vectors using this?
It's a minor matter to show that for a nonempty set $X$ the set of functions $\{f\mid f:X\to \Bbb R \}$ is a vector space under the operations $(f+g)(x):=f(x)+g(x)$ and $(\lambda f)(x):=\lambda f(x)$. Thus these functions qualify as vectors. (Actually $\Bbb R$ can be replaced with any field.)
What about vectors being "lists of numbers"?
The most relevant theorem is this:
This gives us a corollary
What's the connection? The isomorphism in the corollary is given by writing out the cofficients for an element of $V$, and then mapping that element to the list of coefficients in $\bigoplus_{i\in I} F$. So in this sense, it's true that every $F$-vector space looks the same as a list of elements of $F$.
What about vectors as "arrows"?
This is a geometric interpretation of vectors. When working over the real numbers (or any ordered field for that matter) and in two or three dimensions, it is very useful to think of vectors this way.
But this is not really the essence of what a vector is. After you go to even higher dimensions, perhaps infinitely many, the usefulness of the arrow becomes less clear. And also as you move to unordered fields, say $\Bbb C$ or even finite fields, there is no notion of "direction" along the vector, so usefulness diminishes there as well. So the direction-magnitude picture of vectors is a useful picture of real vector spaces, but it is not so successful for general vector spaces.
Now for your bullet points
$^\ast$ Caution: Physicists and engineers talk about vectors and vector spaces differently. The suggested duplicate discusses this: How to think of a function as a vector?