Difference between subspace and subset

Question

Can you give the definition of subspace and subset of $\mathbb{R}^n$ and how can I determine their dimension?

Answer 1

A subset of $\mathbb R^n$ is any set that contains only elements of $\mathbb R^n$. For example, $\{x_0\}$ is a subset of $\mathbb R^n$ if $x_0$ is an element of $\mathbb R^n$. Another example is the set $S=\{x\in\mathbb R^n, ||x||=1\}$.

A subspace, on the other hand, is any subset of $\mathbb R^n$ which is also a vector space over $\mathbb R$. That means that for every $x,y\in S$ and $\alpha\in\mathbb R$, $x+y$ and $\alpha\cdot x$ must also be elements of $S$ in order for $S$ to be a subspace. In our two cases above, $\{x_0\}$ is only a subspace if $x_0=0$, and $S$ is not a subspace.

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EDIT:

An example of a subspace of $\mathbb R^n$ is the set $S_{x_0}=\{\alpha x_0| \alpha\in \mathbb R\}$ which is, if $x_0\neq 0,$ a one-dimensional subspace of $\mathbb R^n$ (and, if $n\neq 1$, the dimesion of $S_{x_0}$ is not $n$).