Mathematical Tools for Physics

6—Vector Spaces 161

6.6 For the vectors in three dimensions,

~v 1 =ˆx+y, ~vˆ 2 =yˆ+ˆz, ~v 3 =zˆ+ˆx

use the Gram-Schmidt procedure to construct an orthonormal basis starting from~v 1.

6.7 For the vector space of polynomials inx, use the scalar product defined as

〈

f,g

〉

=

∫ 1

− 1

dxf(x)*g(x)

(Everything is real here, so the complex conjugation won’t matter.) Start from the vectors

~v 1 = 1, ~v 2 =x, ~v 3 =x^2 , ~v 4 =x^3

and use the Gram-Schmidt procedure to construct an orthonormal basis starting from~v 1. Compare these results
to the results of section4.9. [These polynomials appear in the study of electric potentials and in the study of
angular momentum in quantum mechanics: Legendre polynomials.]

6.8 Repeat the previous problem, but use a different scalar product:

〈

f,g

〉

=

∫∞

−∞

dxe−x

2

f(x)*g(x)

[These polynomials appear in the study of the harmonic oscillator in quantum mechanics and in the study of
certain waves in the upper atmosphere. With a conventional normalization they are called Hermite polynomials.]

6.9 Consider the set of all polynomials inxhaving degree≤N. Show that this is a vector space and find its
dimension.

6.10 Consider the set of all polynomials inxhaving degree≤N and only even powers. Show that this is a
vector space and find its dimension. What about odd powers only?