Consider the basis functions X1 t Q X2t bt C Find the coe

Consider the basis functions X_1 (t) = Q, X_2(t) = bt + C. Find the coefficients a, b, and c so that these functions are orthonormal on the interval 0

Solution

1a) x₁(t)x₂(t)dt = 0 => a(bt +c)dt from 0 to 1 is 0

that is, [abt²/2 + ct] from 0 to 1 = 0 or ab/2 + c = 0 => ab =-2c

Coefficients satisfy ab = -2c

1b) Just the upper limit changes, thus ab *2²/2 + 2c = 0 or 2ab + 2c = 0 => ab = -c