# The meaning of notation like $f\colon \mathbb R^2 \to \mathbb R$, $x \in \mathbb R^n$, and $x \in \mathbb R$.

I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation.

I have tried searching on the web but have had difficulty in finding something which properly explains the meaning of notation like

$$f: \Bbb{R^2} \to \Bbb{R}$$ or the difference between $x\in \Bbb{R^n}$ and $x \in \Bbb{R}$

I can basically read these, and know the literal pronounciation of the symbols, but have no idea what they actually mean.

The first one would be $f$ maps $\Bbb{R^2}$ to $\Bbb{R}$. What does this mean exactly?

Is it saying that on an $(x,y)$ plane, the function $f$ returns a single number? E.g $f(x) = 3x^2$ $f(1) = 3$?

Is the second one saying that $x$ is an element of a vector space with $n$ elements $(ax_1, bx_2,....,a_nx_n)$, whereas the first one is saying that $x$ is just some real number?

I would really appreciate if someone could help me with this, either explaining it or referring me to a nice book that is appropriate at a beginner level. Further more does this type of notation have any specific name?

When we say that $x \in \mathbb R$, we mean that $x$ is simply a (one-dimensional) scalar that happens to be a real number. For example, we might have $x = -2$ or $x = 42$.

On the other hand, when we say that $\vec x \in \mathbb R^2$, we mean that $\vec x$ is a two-dimensional vector whose two components are both real numbers. In other words, $\vec x$ is an ordered pair in the Cartesian plane that has the form $(x_1, x_2)$, where $x_1,x_2 \in \mathbb R$. For example, we might have $\vec x = (-1, 7)$ or $\vec x = (\pi, 2.54)$.

When we define a function $f\colon \mathbb R^2 \to \mathbb R$, we mean that $f$ maps each ordered pair (which contains two numbers as input) to a single number (as output). For example, we could define such a mapping by: $$f((x_1, x_2)) = 2x_1 + 3x_2$$ so that in this case, $f$ would map $\vec x = (-1, 7)$ to $2(-1) + 3(7) = 19$. [Usually, we like to abuse this notation a bit and drop one of the pairs in the double pair of brackets so that we just write it as $f(x_1, x_2)$ instead.] You can visualize this three dimensionally by plotting the point $(-1, 7, 19)$ and all other triples of the form $(x_1, x_2, f(x_1, x_2))$. Using Wolfram|Alpha, we can plot this to obtain a plane passing through the origin $(0, 0, 0)$. In fact, this is the subspace consisting of all possible linear combinations of the basis vectors $(1, 0, 2)$ and $(0, 1, 3)$: • I am in tears. This is such a simple explanation, I wish maths authors could explain things in layman terms for students like myself. Thank you so much, you are an incredibly generous person for helping me with something which is probably trivial to you! Jan 6, 2015 at 7:25

What exactly does this mapping mean? Well, simply put it means that there is a rule, which we shall define by our function $f$, that returns a value in the codomain for every value in the domain.

So let's define these as well. The domain is the set of objects for which our function takes an argument. But we need to make sense of this in some way. So let's consider some polynomial equation, $P$. First, let $P$ be a function

$$P:\mathbb{R}^2\rightarrow \mathbb{R}$$

It is useful to understand what values your function will take as an argument and which values your function returns as. We say that the function takes argument in the domain. This will always be the first set defined before the arrow in our above function notation. In our specific case, the domain is $\mathbb{R}^2$ which is the set of all ordered pairs $(x,y)$ where $x$ and $y$ are values of $\mathbb{R}$.

The function will return values in of the codomain. This is the set immediately following the arrow. Here, the set is of all real numbers.

Let's define our polynomial as taking the ordered pair $(x,y)$ to the single value $x^2+y^2$. We have notation for this as well; we say $(x,y)\mapsto x^2+y^2$ or the value $(x,y)$ gets mapped to $x^2+y^2$.

To me, it seems a lot of the confusion is coming from what the sets $\mathbb{R}^n$ are for each $n$ a natural number. So let's go over this as well.

It is certainly confusing when we say both $x\in \mathbb{R}^n$ and $x\in \mathbb{R}$. First, we need to make something clear. This is that the element $x$ is to be considered an element of the set. This does not mean $x$ is a variable like usual. It means exactly as you say, $x$ defines a point in the vector space $\mathbb{R}^n$ (or $x$ is an element of the set $\mathbb{R}^n$. This means that $x=(x_1,...,x_n)$ for some $n$, and each $x_i$ is a real number; an element of the set $\mathbb{R}$. Note $n$ can be one).

Perhaps I could provide a better visualization than the one taught with orthogonal real lines as is the norm. Consider the space $\mathbb{R}^n$. As a set, we can imagine this as $n$ copies of the real line $\mathbb{R}$. The real line is just an infinitely long continuous line. Now, for each copy of the real line, there is a point on the line that defines the first coordinate of our point $x$. So for the first line in $\mathbb{R}\times...\times \mathbb{R}$ ($n$ copies) corresponds to the value $x_1$ in $x=(x_1,...,x_n)$. We do this for each line and finally achieve a point in the space $\mathbb{R}^n$, namely, $x=(x_1,x_2,...,x_n)$!

Hopefully this clears some things up but, I encourage you to ask questions if I'm confusing or unclear or more issues arise.

Furthermore, the wiki page on functions is pretty insightful to the defintion. I suggest looking through it if confusion persists!

• Eoin you've been incredible helpful. Everything you have said has improved my understanding, I thank you for that. I have only one other question. I have never seen such an intuitive definition of these terms and notations in any textbook so far (although I haven't read that many either). Is this something you developed and learned yourself, or something that you came across in your readings? Jan 6, 2015 at 7:29
• @yemens: You must have had bad luck with your choice of textbooks! These things are usually explained very early in the text. Jan 6, 2015 at 9:44
• @HansLundmark would you care to recommend any books on linear algebra that cover these fundamentals? Jan 6, 2015 at 10:38
• @yemens I'm glad. No, I did not develop this idea myself. Nor did I find it in my readings. Although, most texts do provide pretty adequate foundations for what is to be explored. I find it more difficult to show explicitly these intuitions in an elementary class. Part of maths isn't to follow but to discover. So when new ideas crop up and enlighten a situation it is indeed a good thing. In this sense, most texts want to teach in a heuristic way (e.g. Herstein) instead of formal way (e.g. Bourbaki).
– Eoin
Jan 6, 2015 at 22:26
• @yemens I do not know very many texts on elementary linear algebra. However, if you want to read through some preliminaries I do suggest Topics in Algebra by Herstein. Sometimes you find what you're looking for in other subjects instead of the one you wanted to study originially. I wouldn't read much else in Herstein if you're not interested in Abstract Algebra though! (exploring the internet may help). Sorry if I can not recommend very many other linear algebra books. I am not too fond of the subject myself. Try asking your professor (if possible) for recommendations.
– Eoin
Jan 6, 2015 at 22:30