Dimension of a vector space/ subspace with a finite basis

Question

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?

Answer 1

No, that the number of elements in the vector gives you the dimension of the space where your subspace lies. The dimension of the subspace is given by the number of linearly independent vectors in your generating set. To find those vectors, (called basis) you can write the vectors in a matrix form and convert to REF using elementary row operation along with some zero rows in the bottom. The number of non zero rows(read rank) will give you the dimension of the subspace.

Answer 2

"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.