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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!

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  • $\begingroup$ This question has been asked more than once. The most convincing argument for me is that vectors are elements of vector spaces, and the functions we're talking about form a vector space so... they're vectors. This requires you to think of vector spaces abstractly, of course. It would pay off to start thinking of them that way. $\endgroup$ – rschwieb Aug 18 '14 at 16:57
  • $\begingroup$ What about yourself? Can you easily associate functions with vectors? $\endgroup$ – Kaster Aug 18 '14 at 16:58
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    $\begingroup$ Try going the other way: a vector in $\mathbb{R}^n$ is a function from $\{ 1,\dots,n \}$ to $\mathbb{R}$. Purely from the perspective of linear structure, function spaces with infinite domains work just the same. That is, we define operations pointwise, like: $(f+g)(x)=f(x)+g(x)$, $(c*f)(x)=c*f(x)$, etc. The big difference is not in the linear structure, but in the norm/metric/topological structure. $\endgroup$ – Ian Aug 18 '14 at 16:58
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    $\begingroup$ I guess the main thing we can say to help you now is "you're too fixated on the notation of vectors: you think they're lists of numbers with such and such notation and you're distracted from the bigger idea. Take a look at the definition of a vector space, and learn how some sets of functions form vector spaces." $\endgroup$ – rschwieb Aug 18 '14 at 17:04
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    $\begingroup$ Unfortunately, general vectors are abstractions of euclidean vectors (those point arrow thingys in $\mathbb{R}^2$ and $\mathbb{R}^3$). Most of the properties that your peers think of being associated with vectors are actually just properties of euclidean vectors (euclidean vectors are just a special case of a vector space just as $\mathbb{R}$ (which has operations other than + and $\times$) is just a special case of a Field). $\endgroup$ – John Joy Aug 18 '14 at 17:52
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Let me start by giving a unified viewpoint, and then I'll reconcile the other things you heard with it.

A vector is an element of a vector space*.

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:

Every vector space has a basis $\{b_i\mid i\in I\}$ for some index set $I$.

This gives us a corollary

Every vector space $V$ over a field $F$ is isomorphic to a direct sum $\bigoplus_{i\in I} F$ for some index set $I$.

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

  1. "Vectors are [...] basically pointy arrows written with brackets or an arrow on top..." That is notation, yes, and I think we cleared this concept up :)
  2. "never drawn as an arrow" Yes, because the vector has too many dimensions for an arrow to be a very helpful picture. "doesn't satisfy some axioms of vector space" In general this is just incorrect. Many sets of functions satisfy the axioms of vector spaces.
  3. "You can do [lots of other stuff] with functions [that you can't do with vectors]" Well, there is no rule about vectors that they can't also be other things with extra special abilities! The set of $n\times n$ square matrices are vectors in a vector space too, but being square does not prevent you from being a vector!
  4. "No authority has ever said functions are vectors" This is just misinformed. The entire field of functional analysis concerns itself with vector spaces of functions.
  5. "Yes, functions are sometimes contained in brackets, but that is just a vector of functions, not a function" This is a slightly sticky question because the tuples involved are different (and aren't different.) This is another good reason to stop thinking of vectors only as "tuples of numbers." You can have tuples of functions, and yes they could still be thought of as vectors in the product space $V\times V$.

$^\ast$ 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?

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There is another way function could be viewed as a vector. For practical calculations, plotting, etc. functions are sometimes discretized, sufficiently finely. For example, $x\mapsto x^2$, $x\in[0,3]$ can be represented as its values on the nodes 0,1,2,3 as a 4-dimensional vector $(0,1,4,9)^T$.

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Just a food for thought for your peers, consider $f:\mathbf{R}^{n}\rightarrow\mathbf{R};f_{\vec{u}}(\vec{v})=\vec{u}\cdot\vec{v}$ is uses a vector to create a function which acts as a simple linear combination of the elements of $\vec{u}$. This is a just a special case of a linear transformation, $f(\vec{u})=A\cdot\vec{u}$ , where in this case $A$ is $n\times 1$.

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