# Quick Vector Space Question

I am working on a practice exam and one of the questions is:

Always True or False: The set of real numbers $\mathbb{R}$, is a vector subspace of $\mathbb{R}^{1 \times 1}$

I don't really know how to approach this problem because I don't know what it means by $\mathbb{R}^{1 \times 1}$. I've tried searching for this on the internet and even in my textbook but I can't seem to find an example with the cross notation in the vector space.

Any help would be great.

## 1 Answer

$\mathbb{R}^{1\times 1}$ is the set of $1\times 1$ matrices, with real entries. $$\mathbb{R}^{1\times 1}=\{\left[x\right]:x\in \mathbb{R}\}$$

This is not $\mathbb{R}$, since $\mathbb{R}$ are numbers without any brackets. In fact, the two have no elements in common at all.

There is, however, a natural isomorphism between them $f:\mathbb{R}\to \mathbb{R}^{1\times 1}$ given by $f:x\mapsto [x]$.

In short, this is a very hair-splitty type of question, that tests technical parts of a definition.

• Ahh, I thought $M_{1 \times 1}$ represented that set. Is there a difference between $\mathbb{R}^{1 \times 1}$ and $M_{1 \times 1}$? – LinearAlgebraStudent Feb 18 '15 at 23:27
• There isn't really standard notation for matrix spaces. I've seen both of those used. – vadim123 Feb 18 '15 at 23:27