# What do exponents of $\mathbb{R}$ denote?

An equation in my machine learning class says
$$x= \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_n \\ \end{bmatrix} \in ℝ^{n+1}$$

I'm reading this as "x is the subset of numbers belonging to the set off all real numbers that are the square of the set total plus one" and I don't feel that's correct.

• $ℝ^{2}$ means 2 dimensions, $ℝ^{3}$ means 3 dimensions, $ℝ^{4}$ means 4 dimensions, and so on.
– ndh
Oct 22 '13 at 5:30
• It's a set of all n-tuples of real numbers, i.e. that belong to $\mathbb R$. Oct 22 '13 at 5:30
• x is an element in the set of all vectors of length $n+1$ where each "cell" in the vector has a real number Oct 22 '13 at 5:31

First of all $x$ isn't a subset of $\mathbb{R}^{n+1}$, it is an element of $\mathbb{R}^{n+1}$.
Let $X$ be a fixed set. Then $X^{n+1}$ is the set of ordered $(n+1)$-tuples of elements of $X$. That is, $$X^{n+1} = \{(x_0, x_1, \dots, x_n) \mid x_0, x_1, \dots, x_n \in X\}.$$ In the case where $X = \mathbb{R}$, these are just real $(n+1)$-dimensional vectors that we usually write as columns, as you have done.
For any set $X$, the notation $X^n$ denotes the set of all $n$-tuples of elements of $X$: the lists $(x_1,x_2,\ldots,x_n)$ where each $x_i$ is an (independently chosen) element of $X$. Therefore $\Bbb R^{n+1}$ denotes the set of all $n+1$-tuples of real numbers, which are here written vertically, and indexed starting from$~0$.
The equation simply states that $x$ is a $(n+1) \times 1$ vector. So $\mathbb{R}^{n+1}$ is the space where $(n+1) \times 1$ vectors lie.