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
Best Answer
The differences:
A basis is a subset of the vector space with special properties: it has to span the vector space, and it has to be linearly independent.
The initial set of three elements you gave fails to be linearly independent, but it does span the space you specified. In that case you just call it a generating set.
The dimension of a finite dimensional vector space is a cardinal number: it is the cardinality of a basis (any basis!)
The rank of a linear transformation is the dimension of its image. That is, if you have a linear transformation $f:V\to W$, the rank of $f$ is $\dim(f(V))$. This is the most common usage of the word "rank" in regular linear algebra. I can also imagine some authors unfortunately using "rank" as a synonym for dimension, but hopefully that is not very common.
So the three things refer to each other (dimension is the size of a basis, rank is the dimension of the image) but you can see they are different things.
The vector space you mentioned does indeed have dimension $1$. It is a subspace of a vector space of dimension $3$ ($\mathbb R^3$), but it does not have dimension $3$ itself. Its bases only have $1$ element, but every basis of $\mathbb R^3$ has three elements. The only relationship between the dimension of a vector space $V$ and a subspace $W\subseteq V$ is that $\dim(W)\leq \dim(V)$, which your example demonstrates.