Notation of real numbers

Question

Anyone who can explain me in short Words what these notations means:

$\mathbb{R}^m$ or $\mathbb{R}^{m \times n}$

Please if possible With an example, thank you!

Answer 1

To be more specific than lulu's comment:

$\Bbb R^1 = \Bbb R$, the set of real numbers.

$\Bbb R^2 = \Bbb R \times \Bbb R = \{(x, y) \mid x, y \in \Bbb R\}$, the set of all ordered pairs of real numbers. If you think of the ordered pairs as $x$ and $y$ coordinates, then it can be identified with a plane.

$\Bbb R^3 = \{(x, y, z) \mid x, y, z \in \Bbb R\}$, the set of all ordered triples of real numbers. This can be identified with space.

Etc.

By this same pattern, $\Bbb R^{n\times m}$, is an (ordered) $nm$-tuple: $(x_1, ..., x_{nm})$. But we can consider the entries in groups of $m$ : $$\begin{array}{ccccc}(&x_1,& ..., &x_m\\&x_{m+1},& ...,& x_{2m}\\&x_{2m+1},& ...,& x_{3m}\\&&\vdots\\&x_{(n-1)m+1},& ...,& x_{nm}&)\end{array}$$ reindex them with pairs of indices, drop the commas in favor of spacing, and extend the brackets: $$\begin{pmatrix} x_{11} & \dots & x_{1m}\\x_{21} & \dots & x_{2m}\\& \vdots\\x_{n1} & \dots & x_{nm}\end{pmatrix}$$ and suddenly our $nm$-tuple is an $n \times m$ matrix. So the specific notation $\Bbb R^{n\times m}$ is sometimes used to represent all $n \times m$ real matrices.