What are the types of elements found in a vector space other than n-tuples?

Question

So basically when we talk about vector space, we consider a set of arbitrary elements and some operations in it which constitute it. Most common type of element known is that of n-tuples of say reals. What other varieties of elements exist in a vector space? Can a set of matrix be considered for constituting a vector space?

Answer 1

Partial answer only:

A vector space is a set in which you can add two of its elements, and multiply any element by a “scalar”, subject to certain rules that I won’t mention here. Any of one those elements may be called a vector, if you like.

Standard example: the set of all $n$-tuples of real numbers, where you add them coordinatewise, and multiply by a real scalar $\lambda$ by multiplying each coordinate by that $\lambda$.

Other examples:
$\quad$1. the vector space is the set of all polynomial expressions in a single variable (say $x$), with real coefficients. If you want to call $1-3x+x^2$ a vector, you may, as long as you’ve made it clear that your vector space is some specified set of polynomials (maybe all), satisfying the rules.
$\quad$2. The vector space is $\Bbb R$, the set of real numbers, but the allowed scalars are just the rational numbers $\Bbb Q$. This is an example of an infinite-dimensional vector space over the rationals.
$\quad$3. You may want to talk about the totality of all two-by-two matrices with real entries. This is a vector space over the real field $\Bbb R$, of dimension four. A single matrix will be a vector in this space.
$\quad$4. The set of all possible sequences of real numbers $a_1,a_2,a_3,\cdots\,$. You may think of these as inf-tuples of reals, and any one of them would be a vector in this vector space.
$\quad$5. The set of all real sequences as above, but subject to the condition that $\lim_{i\to\infty}a_i$ exists as a real number. You can add any two of these to get another, as you know, and if you take a convergent sequence $\{a_i\}$ you can multiply every term by a given $\lambda$ to get another convergent sequence, as you also know. So that’s a good vector space, too.