I am a beginner in Linear Algebra, I was reading about introduction to lists, It is defined in the book that list is collection of $n$ elements in a certain order of length $n$, represented as $(a,b,c......n)$. My questions are,
1.Can the elements of the list be functions of $n$ variables. An example would be $(x-y,x+2y,x+7y,...,n\text{th function})$.
2.If so then can the list represent a vector space such that it follows the properties of scalar multiplication and addition required for it to be called a vector space with a certain fixed values of $x$ and $y$.
I request you to point out if there are any mistakes in concept of the question.
You can define a Cartesian product of sets as a set whos members are ordered lists (or lists for short). That is, let $A$ be a non-empty set then we define $$A^{n}:=\underbrace{A\times A\times \dots \times A}_{n}:=\{(a_{1},a_{2},\dots,a_{n}):a_{i}\in A\}.$$ Note here that nothing specifies what the elements of $A$ should be.
(Ofcourse you need not use the same set $A$ for each member in the list. You could write $A\times B$ where $A$ and $B$ are different sets.)
In order to use this set as an underlying set for a vector space: you must choose a field of scalars, define addition and scalar multiplication on $A^{n}$ and make sure the vector space axioms are satisfied.