Thread: gram-schmidt process help

Hi everyone,

i need your help with solving the (ii) choice of the quesiton i attached,thanks for your helps .

by the way i confused how to use (u,v) and p(t),q(t),
i really appreciate if you show me the details of obtaining g1 and g2 i can find g3.

ok, since we can pick any basis vector to start with, i'll choose g₁ = b₁ = 1 (the constant polynomial, not the number) that is: b₁(t) = 1, for all t.

now to find g₂, we have the find the projection of b₂ in the direction of 1. we have to use the given inner product for this, but the general formula is:

proj_g1(b₂) = [(g₁,b₂)/(g₁,g₁)]g₁.

note that g₁(0) = g₁(1) = g₁(2) = 1 (it's a constant function). thus (g₁,g₁) = (1)(1) + (1)(1) + (1)(1) = 3.

the real action comes in when we compute (g₁,b₂). b₂(t) = t, so:

b₂(0) = 0, b₂(1) = 1, b₂(2) = 2.

thus (g₁,b₂) = g₁(0)b₂(0) + g₁(1)b₂(1) + g₁(2)b₂(2) = (1)(0) + (1)(1) + (1)(2) = 3.

hence proj_g1(b₂) = (3/3)(1) = 1 (the constant polynomial, not the number).

to find g₂, we simply subtract the projection of b₂ in the direction of g₁ from b₂:

g₂(t) = b₂(t) - proj_g1(b₂)(t) = t - 1.

let's verify that g₂ is orthogonal to g₁:

(g₁,g₂) = (1,t-1) = g₁(0)g₂(0) + g₁(1)g₂(1) + g₁(2)g₂(2) = (1)(-1) + (1)(0) + (1)(1) = 0. see?

finding g₃ will be the same, except now you have to find 2 projections, and subtract them both from b₃(t) = t².