Normalizing function

Question

Assignment:

Given the set of functions:

f₁(t) = A₁e^-t

f₂(t) = A₂e^-2t

Defined on the interval (0, ∞).

(a) Find A₁ such that f₁(t) is normalized to unity on (0, ∞).  Call this function Φ₁(t).

(b) Find B such that Φ₁(t) and f₂(t) - BΦ₁(t) are orthogonal on (0, ∞).  Normalize this new function and call it Φ₂(t).

(c) Do this for the third function, e^-3t.  That is, chooses C and D such that f₃(t) - CΦ₂(t) - DΦ₁(t) is orthogonal to both Φ₁(t) and Φ₂(t).  Normalize it and call it Φ₃(t).  Comment on the feasibility of continuing this procedure.

Provide complete and step by step solution for the question and show calculations and use formulas.