in linear algebra, we played with the idea of multiple variables. these took the forms of linear systems like: 2x + 3y + 5z = 0. if we remember, these equations made up lines, planes, and hyperplanes through the origin- all tied together with their main identifer: linearity. but not every function is linear! so when we have a nonlinear function, what does that mean? what do they look like and how do we work with them?

first, let's reminisce on dimension. the number of basis vectors we have defines our dimension- we'll only worry about the reals for now, so our dimensions will be notated Rn (if we wanted to work with complex numbers, we could say Cn!). the basis vectors for R2 are x and y, basis vectors for R3 are x, y, and z, and so on. we can also name these basis vectors arbitrarily, like when we have 27 dimensional space and we've run out of letters, it will be much easier to write our basis as x1, x2, x3.... x27. these basis vectors are linearly independent and are all zeros except for a 1 in the slot of what index they are in the list (ex: x1 = [1,0,0] for R3).

now let's say we have some function here: x^2 + y^2 = 1. hopefully we recognize this guy, it's a circle with radius 1! right now, our goal is to be able to understand where this function lives. dimension is a quality of a space, and functions live in spaces. so how do we connect the two? a good guess would be through the variables, we have two variables, so maybe this function lives in R2? it's a great guess, but i'll return that guess with a little note attached: do we really have two variables?

i know you might be thinking, damie, of course we have two variables, you wrote x and y! and you're right, i did! but that equals sign shakes things up a bit. what we're doing in this function is defining a relationship. we're saying: we've got this x, and this y, and this is the link between these two. through this function, i can write x(y) or y(x)! that means