I'm having problems with BASIS questions, please help explain

I don't know understand what is BASIS in vector. I try to read it on the book but still.. :(

I want to solve this question but I don't know how to begin with. Someone please help explain.

Finding a basis for the set of the function spanned by x²-1, x²+1, x, and x+1

Hey angelme.

A basis is just a collect of vectors that describe the whole space you are dealing with. In geometry we usually write any three dimensional vector as V = ai + bj + ck where a,b,c are real numbers and i,j,k are vectors corresponding to the x,y,z axis in 3D geometry. They don't have to be these specific vectors but this is a common basis: as long you can construct any vector in the space using a linear combination of the basis vectors, then the basis vectors constitute a basis.

Instead of having a vector in <x,y,z> you have a polynomial in <x^2,x,1> where you have basis vectors x,y,z such that X = ax + by + cz describes all possible vectors X in your space.

Now you need to identify a basis for the polynomial and the first step is to think of the x^2 terms independent to x terms and x terms independent to constant terms: in fact all three will be independent from each other and then think of these as vectors <x,y,z> instead of <x^2,x,1>

So the basis for the set of the function spanned by x^2 -1, x^2 +1, x, and x+1 is (x^2,x,1) simple as that ?

let's write our functions as vectors with coordinates in the basis: {1,x,x²} (we can do this because all 4 functions are polynomials of degree less than or equal to 2).

since x² - 1 = (-1)(1) + (0)(x) + (1)(x²), in this basis, x² - 1 has coordinates: (-1,0,1).

similarly, in this basis, x² + 1 has coordinates: (1,0,1).

x has coordinates: (0,1,0), and x+1 has coordinates: (1,1,0).

so now can ask: what is the space spanned by: S = {(-1,0,1),(1,0,1),(0,1,0),(1,1,0)}? once we've found that, we can find a basis for it, by choosing a maximal linearly independent subset of S.

pro tip: since we have just 3 coordinates, our maximum possible dimension (basis size) is 3. can you prove we need at least 3?

(your answer is correct, but do you know WHY?)