Polynomial Orthogonality I study Polynomial Orthogonality and in connection to Linear Algebra I visualize the concept this way :

*

*$f(x)$ and $g(x)$ are polynomials

*Any polynomial $p(x)$ in the interval $[a,b]$ is a vector with infinite coordinates as $[a,b]$ contains an infinite amount of numbers $[x_0,x_1,\dots,x_n]$ and it's coordinates are $(p(x_0),p(x_1),\dots,p(x_n))$

*The inner product of $f(x)$ and $g(x)$ is equal to : $f(x_0)*g(x_0)+f(x_1)*g(x_1)+\dots+f(x_n)*g(x_n)$

*The process above can also be expressed as : $\int_{x_0}^{x_n} f(x)*g(x) \,dx$
I got 3 questions though :

*

*Is there a better/more accurate way to visualize the concept described above?

*In most cases/examples I've seen writers limit themselves to the interval $[-1,1]$.Why is that? I tried some plotting and integration in Desmos an indeed in most cases only the interval $[-1,1]$ takes the inner product to $0$.

*Besides orthogonality and in connection to Linear Algebra is there a way to show that two polynomials over an interval $[a,b]$ are as vectors linearly indpendent?

 A: It's not how it works.
First, the set of polynomials with coefficients in a field $\Bbb K$ (for instance $\Bbb K=\Bbb R$) is a vector space over the field $\Bbb K$. A natural basis of this vector space is $(1,X,X^2,\dots)$. It's also possible to consider only polynomials of degree $\le n$, which is a finite dimensional vector space, of dimension $n+1$.
Given an inner product of this vector space, it's then possible, using Gram-Schmidt orthogonalization, to build an orthogonal basis.
So, it remains to find an inner product. A common way (but not the only one), is to define
$$\left<P,Q\right>=\int_a^bP(x)Q(x)w(x)dx$$
Where $a,b$ are fixed but arbitrary, and possibly infinite, and $w$ is a weight function. Of course, the integral must be convergent for any polynomial $P,Q$.
For instance, with $w=1$ and $[a,b]=[-1,1]$, you get Legendre polynomials. Examples on an unbounded interval include Laguerre polynomials and Hermite polynomials.
More on this on Wikipedia:

*

*Orthogonal polynomials

*Classical orthogonal polynomials

To answer your last question: two polynomials are linearly independent if their quotient is not a constant. More generally, it's possible to check the linear independance of a set of $n$ polynomials, using for instance a Gram determinant.
For this, you need an inner product: you may use one of those described above. It's also possible to use the fact that the vector space $\Bbb R_n[X]$polynomials of degree $\le n$ is isomorphic to $\Bbb R^{n+1}$ (using the values of these polynomials at $n+1$ points, thanks to the linearity of Lagrange interpolation), then use an inner product on $\Bbb R^{n+1}$.
