What does "$ℝ^{4\times10}$" mean

Question

I was given this statement:

where $W\in\mathbb R^{4\times 10}$ and $\vec{b}\in\mathbb R^4$.

Now, I understand that $ℝ^{4}$ means that the set of real numbers in four dimensions. However, I am not sure what $ℝ^{4\times10}$ mean?

Answer 1

The notation

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

is sometimes used to refer to the vector space of two-dimensional arrays of real numbers consisting of $m$ rows and $n$ columns.

It is rather common to identify this with the space of $m \times n$ real-valued matrices.

There is a nice coincidence of notation, since as abstract vector spaces there is an isomorphism

$$ \mathbb{R}^{m \times n} \cong \mathbb{R}^{mn} $$

obtained by rearranging the entries into a one-dimensional array.