I think an interesting point to note is that we can also apply Gram-Schmidt to functions, as long as we have a well defined inner product. For example, for the set ${1, x, x^2,...}$ and defining $<f,g> = \int_{-1}^{1} f(x)g(x) dx$, the Gram-Schmidt process generates the Legendre polynomials - pretty cool! Also, by changing the definition of our inner product, we can get other orthogonal polynomials such as Hermite polynomials.
keenan
Very nice! We will talk more about polynomial bases when we look at spline interpolation (later in the semester).
stutiRastogi
I don't think the order should matter when it comes to bases, because they are just a set of n vectors. In Gram-Schmidt algorithm, we can choose any one to be the "1st vector" for the algorithm.
wumengjz
I think the order of bases does matter in 3D.
Specifically, the normals are different between two 3D triangles with bases (e1,e2) and (e2,e1), respectively. The normal is the cross product of two bases, which follows the right-hand rule.
@wumengjz Yes, exactly: the order of bases can influence the direction of the normal, and this orientation has important implications for practical computation (e.g., for shading a surface, or computing a physical/geometric flow). Orientation will continue to show up in several places throughout the semester.
I think an interesting point to note is that we can also apply Gram-Schmidt to functions, as long as we have a well defined inner product. For example, for the set ${1, x, x^2,...}$ and defining $<f,g> = \int_{-1}^{1} f(x)g(x) dx$, the Gram-Schmidt process generates the Legendre polynomials - pretty cool! Also, by changing the definition of our inner product, we can get other orthogonal polynomials such as Hermite polynomials.
Very nice! We will talk more about polynomial bases when we look at spline interpolation (later in the semester).
I don't think the order should matter when it comes to bases, because they are just a set of n vectors. In Gram-Schmidt algorithm, we can choose any one to be the "1st vector" for the algorithm.
I think the order of bases does matter in 3D. Specifically, the normals are different between two 3D triangles with bases (e1,e2) and (e2,e1), respectively. The normal is the cross product of two bases, which follows the right-hand rule.
Reference: https://en.wikipedia.org/wiki/Cross_product
@wumengjz Yes, exactly: the order of bases can influence the direction of the normal, and this orientation has important implications for practical computation (e.g., for shading a surface, or computing a physical/geometric flow). Orientation will continue to show up in several places throughout the semester.