Dimension of a vector space/ subspace with a finite basis Is the dimension of a vector space/subspace with a finite basis always the same as the number of elements in each vector and if so how can I derive that from the definition of a dimension?

"Elements" of a vector is not something that makes sense in an abstract vector space. It does make sense when considering vectors in $\Bbb{R}^n$ (or more generally, $\Bbb{F}^n$ for some field $\Bbb{F}$), but not for more general vectors (e.g. functions).
Even if we restrict ourselves to vectors in $\Bbb{R}^n$, the number of elements in a vector doesn't correspond necessarily to the dimension of the space. For example, if we consider the vector space
$$V = \{(x, y, z) \in \Bbb{R}^3 : x + y + z = 0\},$$
then each vector has exactly $3$ elements, but the space is $2$-dimensional (though, of course, it is a subspace of the $3$-dimensional space $\Bbb{R}^3$).
What makes more sense, instead of "elements", is "coordinates". Given any finite-dimensional space $V$ (i.e. one with a finite basis $B$), we can express vectors in $V$ as unique linear combinations of vectors in $B$. We can describe a vector $v \in V$ completely by the coefficients in the unique linear combination of vectors in $B$ that forms $v$. Since every basis has the same number of elements, every coordinate system has the same number of coordinates. This number is what we call the dimension of $V$.
So, in summary, "number of elements" of a vector doesn't make sense in general, but "number of coordinates" does make sense and is the very definition of dimension.
