vector spaces, linear independence and functions

So here is the question. This is part of an essay. this is in the context also of the four fundamental subspaces. Obviously i'm not asking for essays but im just confused on this as a whole:

elaborate on how sets of functions and sets of matrices (say 2x2 matrices) can satisfy the definition of a vector space. Discuss what might change when the vectors themselves are not n-tuples. what would it mean for a set of 5 matrices to be considered linearly independent. What about a set of 4 functions?



im confused on the difference of functions and matrices they are talking about here. are they one in the same?? so for a vector space, it must be a subspace of that vector space? so when the vectors are not n-tuples (does that mean not a square matrix?), so does the space of the vectors change as well? im sorry if im not making sense im really confused here we have a teacher that doesnt make much sense and speaks poor english so any help would be great!!!

"Vectors" can be anything at all. For example, consider the set of polynomials with real-valued coefficients. Is it a vector space over the reals? Check the axioms. What about polynomials with integer coefficients? (No, they are not closed under scalar multiplication).

Now, what does it mean for a set of polynomials (remember, just a kind of function) to be linearly independent? It's exactly the same definition as for n-tuples. If there's a collection of scalars, not all zero, one for each polynomial, such that the sum of each scalar times its polynomial is zero (in this case, zero means the zero polynomial, again a function!), then they are linearly dependent.

The point of this exercise is to get you to realize how general the vector space axioms are. Tuples of real numbers are the first vector space everyone sees, but the axioms can be satisfied by many, many kinds of objects. Just figure out what your scalars are, what the zero object is, and check the axioms carefully.

ok, i've been working on this, but i have a more specific question: what does it mean for a vector to not be an n-tuple? I somewhat understand the n-tuple concept, that a vector has n values in it.

A vector is just an element in a vector space. Tuples of field elements are just the first example we usually see. Continuous real-valued functions that are zero at zero form a vector space over the real numbers, too (check that they can be added and multiplied by scalars).

so a continuous function that starts at zero is an example of a vector that is not an n tuple?

ok i understand. so the question "what might change when the vectors themselves are not n-tuples" is really asking what might change when the vectors are continuous.

Again, that's an example of what might change. As another example, the sequences of real numbers form a real vector space, and the elements look almost like tuples (but they're infinite). There are many, many different kinds of objects with vector space structures.