The number of elements in any basis of a vector space is called the dimension of Vector space.
In the Vector space F^n(F) and Null Space; the dimension of Vector Space and the number of Vectors in the Vector Space are distinct.
Does the number of elements in the Vector Space and the Dimension of Vector Space always distinct???
If not,then in which case number of elements in Vector Space and the Dimension of Vector Space both are equal??
We focus on finite-dimensional cases, where we can prove without much effort the dimension and the number of elements never agree.
First suppose we work over an infinite field $\mathbb{F}$ (e.g.$\mathbb{Q}$,$\mathbb{R}$,$\mathbb{C}$). In this case, any finite-dimensional vector space over $\mathbb{F}$ has infinitely many elements. (As an example, we may consider $\mathbb{R}$ itself as a $1$-dimensional vector space over $\mathbb{R}$, which has infinitely many elements. )
Then suppose $\mathbb{F}$ is a finite field of $q$ elements. By the definition of a field, we have $q>1$. It's clear that an $n$-dimensional vector space over $\mathbb{F}$ has $q^n$ elements. So again, we conclude $q^n > n$.
If we do not restrict ourselves to finite-dimensional vector spaces, things can be subtler.