Example 1

P2P2 = {all quadratic polynomials}. Let b0 ⁢t=1 b0 t 1 , b1 ⁢t=t b1 t t , b2 ⁢t=t2 b2 t t 2 .

b0 ⁢t b1 ⁢t b2 ⁢t b0 t b1 t b2 t span P2P2, i.e. you can write any f⁢t∈ P2 f t P2 as f⁢t= α0 ⁢ b0 ⁢t+ α1 ⁢ b1 ⁢t+ α2 ⁢ b2 ⁢t f t α0 b0 t α1 b1 t α2 b2 t for some αi ∈R αi .

f⁢t=t2−3⁢t−4 f t t 2 3 t 4

Alternate basis b0 ⁢t b1 ⁢t b2 ⁢t=1t12⁢(3⁢t2−1) b0 t b1 t b2 t 1 t 1 2 3 t 2 1 write f⁢t f t in terms of this new basis d0 ⁢t= b0 ⁢t d0 t b0 t , d1 ⁢t= b1 ⁢t d1 t b1 t , d2 ⁢t=32⁢ b2 ⁢t−12⁢ b0 ⁢t d2 t 3 2 b2 t 1 2 b0 t . f⁢t=t2−3⁢t−4=4⁢ b0 ⁢t−3⁢ b1 ⁢t+ b2 ⁢t f t t 2 3 t 4 4 b0 t 3 b1 t b2 t f⁢t= β0 ⁢ d0 ⁢t+ β1 ⁢ d1 ⁢t+ β2 ⁢ d2 ⁢t= β0 ⁢ b0 ⁢t+ β1 ⁢ b1 ⁢t+ β2 ⁢(32⁢ b2 ⁢t−12⁢ b0 ⁢t) f t β0 d0 t β1 d1 t β2 d2 t β0 b0 t β1 b1 t β2 3 2 b2 t 1 2 b0 t f⁢t= β0 ⁢ b0 ⁢t+ β1 ⁢ b1 ⁢t+32⁢ β2 ⁢ b2 ⁢t f t β0 1 2 b0 t β1 b1 t 3 2 β2 b2 t so β0 −12=4 β0 1 2 4 β1 =-3 β1 -3 32⁢ β2 =1 3 2 β2 1 then we get f⁢t=4.5⁢ d0 ⁢t−3⁢ d1 ⁢t+23⁢ d2 ⁢t f t 4.5 d0 t 3 d1 t 2 3 d2 t

Example 2

ei⁢ ω0 ⁢n⁢t|n=−∞∞ n ω0 n t is a basis for L2 ⁢0T L2 0 T , T=2⁢π ω0 T 2 ω0 , f⁢t=∑n Cn ⁢ei⁢ ω0 ⁢n⁢t f t n Cn ω0 n t .

We calculate the expansion coefficients with

Infinite-dimensional spaces are hard to visualize. We can get a handle on the intuition by recognizing they share many of the same mathematical properties with finite dimensional spaces. Many concepts apply to both (like "basis expansion"). Some don't (change of basis isn't a nice matrix formula).