When I have a vector space spanned by a set of vectors:$$\mathbf{V}=\mathrm{span}\{(a,b,c,d)^T,(d,e,f,g)^T,(2a,2b,2c,2d)^T\},$$How can I write $\mathbf{V}$ in terms of a subset to another vector space? In this case $\mathbf{V}$ is a subspace of dimension 2, but is it true that it is also a subspace of $\mathbb{R}^3$ and $\mathbb{R}^4$, i.e. $\mathbf{V}\subset\mathbb{R}^3$, $\mathbf{V}\subset\mathbb{R}^4$?
My confusion is that when I write $\mathbf{V} \subset \mathbb{R}^3$, does it also imply the all of the element in $\mathbf{V}$ are triples?
Yes. $V\subset\mathbb R^4$. But $V\not\subset\mathbb R^3$, here.
Though $V$ is $2$-dimensional, it is a subspace of $\mathbb R^4$. Vectors in $\mathbb R^4$ are expressed as $4$-tuples.
Whereas $\mathbb R^3$ "sits" in $\mathbb R^4$ in a fairly obvious way, we cannot say, for instance, that any $2$-dimensional subspace of $\mathbb R^4$ is a subspace of $\mathbb R^3$. If the fourth coordinate is $0$ for all elements of $V$, we can say that, in a sense, $V\subset \mathbb R^3$. But here we would be thinking of $\mathbb R^3$ as the set of $4$-tuples with fourth coordinate $0$. Note there are other $"\mathbb R^3"$s in $\mathbb R^4$ (let each of the other $3$ coordinates be $0$).
Plus there are lots of other there $3$-dimensional subspaces (one corresponding to each normal direction).
Btw, all $n$-dimensional vector spaces are isomorphic.
All this can take some getting used to.