Why is that the number of coordinates has to be greater than the dimension of a given vector space?
Say there is a linear transformation $\textsf{T}:\textsf{V}\to\textsf{W}$ and $\textrm{dim}(\textsf{V})=2, \textrm{dim}(\textsf{W})=3.$ So I have to use at least $3$ coordinates to describe both vector space?
When I read $\beta_{\textsf{V}}=\{v_1,v_2\}$, which is a basis of $\textsf{V}$ and that's ok for me to understand: it's a linearly independent subset of $\textsf{V}$ which spanning $\textsf{V}$. But it becomes quite confusing when you say $\beta_{\textsf{V}}=\{(1,0,0),(0,1,0)\}$. Why use three? not four, five, ...coordinates?
It seems like there is an observer which in a higher dimension to describe the linear transformation?
Note that, in general, if $V\subseteq \mathbb{R^n}$ any element of V is an $n$- dimensional vector but dim(V) can be between $0$ and $n$ depending on the number of vectors in a basis for $V$. You should distinguish the two concepts.