In the Mathematical Methods Physic Eng there is a line in vector algebra chapter
A vector in the three-dimensional space thus required three components to describe fully both direction and its magnitude.
Also there is description of vectors as standard unit vectors
$$ A = a_x \hat i + a_y \hat j + a_z \hat k$$
And we know that $\begin{Bmatrix} \hat i & \hat j & \hat k \end{Bmatrix}$ are basis vectors. From linear algebra knowledge, cardinality of basis vector set is the dimension of vector space.
Does this imply that the number of coordinated points in the cartesian plane is proportional to the dimension of the basis? The basis set has a cardinality of two for a two-dimensional vector and three for a three-dimensional vector.
The dimension of a vector space is the size of a basis (and remember that basis size is invariant). The quote means that in a vector space of dimension $n$, it takes a linear combination of $n$ linearly independent vectors to represent it. But be careful, as you can represent a vector in a subspace with a linear combination of fewer vectors.
To see this, consider the set $S = \operatorname{Span} \{ (1, 0, 0), (0, 1, 0)\}.$ Certainly $S \subset \mathbb{R}^3$ and $\operatorname{dim} S = 2.$ Any vector in $S$ has three entries, but it takes two vectors to represent it inside the subspace.