I think I am a little bit confused with the terms in the title, so I hope you can correct me if I got it wrong... 

$$
\left\lbrace \begin{bmatrix} x\_{1}\\0\\0 \end{bmatrix} : x\_{1} \in \mathbb{R} \right\rbrace
$$ is a vector space. So far so good.

The dimension of the vector space is the number of vectors in the basis.

In class, I wrote down that a basis is

$$
B=(\begin{bmatrix}{1}\\{0}\\{0}\end{bmatrix}, \begin{bmatrix}{0}\\{0}\\{0}\end{bmatrix}, \begin{bmatrix}{0}\\{0}\\{0}\end{bmatrix})
$$

But this does not seem logical to me now, as it should be the smallest number of independent vectors that span the vector space. And shouldn’t that be just 
$
(\begin{bmatrix}{1}\\{0}\\{0}\end{bmatrix})
$

So this is the place where I'm not sure: what is the dimension of the vector space? It is a line in ${\mathbb R}^3$, and it would seem logical that the dimension of a line is $1$.

And what would the rank be? I totally confused myself.

Or is it like the dimension is $3$ and the rank is $1$. so the above solution for the basis would be correct...

Many thanks for your help