University:
University of California, Berkeley
Course:
MATH 54 | Linear Algebra and Differential Equations
Academic year:

2021
Views:

278

Pages:

3
Author:

Kaiden Murphy

a real # Q: is dot product from vector space other than IR^n? Ex: V = Pn(IR), Define = ∫[0 to 1] p(x)q(x)dx i.e. = ∫[0 to 1] x(x+1)dx = ∫[0 to 1] x^2 + x dx => 1/3 x^3 + 1/2 x^2 |[0 to 1] = 1/3 + 1/2 = 5/6 observation: 1/ ∫[0 to 1] p(x)q(x)dx = ∫[0 to 1] q(x)p(x)dx => = 2/ ∫[0 to 1] (p(x) + r(x)) q(x)dx = ∫[0 to 1] p(x)q(x)dx + ∫[0 to 1] r(x)q(x)dx => = + 3/ ∫[0 to 1] (λp(x))q(x)dx = λ ∫[0 to 1] p(x)q(x)dx => <λp(x), q(x)> = λ 4/ ∫[0 to 1] p(x)p(x)dx = ∫[0 to 1] (p(x))^2 dx [downward pointing arrow] ∫[0 to 1] (p(x))^2 dx ≥ 0 [downward pointing arrow] p(x) = 0 ≥ 0 { = 0 => p(x) = 0} Definition: inner product on a real vector space V is a function, to each pair u & v assigns a real # such that. = = + = c ≥ 0, = 0 if and only if u = 0 Ex dot product in R^n = ∫[a to b] p(x)q(x)dx on Pn(IR) Fact: All inner product on IR^n are of the form = u^TAv, where A is an nxn symmetric matrix w/ strictly positive eigenvalues. Example: A = I_n => = u^TI_nv = u^Tv A = [2 0; 0 3] => = u^T[2 0; 0 3]v (standard inner product on IR^n) = 2u_1v_1 + 3u_2v_2 Terminology: ||u|| = √ = Norm of u in V ||u-v|| = Distance between u & v u and v orthogonal in V <=> = 0 Important properties: 1/ Triangle Inequality: ||u+v|| ≤ ||u|| + ||v|| 2/ Cauchy-Schwarz inequality: || ≤ ||u|| ||v|| Key Fact: defined the using inner product transfers to finite dimensional inner product spaces. Ex: V = Pn(IR), = ∫[a to b] p(x)q(x)dx If u = span {1, x^2}, calculate proj_u (x^3) <1,x^2> = ∫[a to b] 1⋅x^2 dx = 1/3 x^3 |[a to b] = 1/3 => apply G.S. u_1 = v_1 u2 = v2 - / u1 = x^2 - ∫[0 to 1] x^2 dx / ∫[0 to 1] 1⋅1 dx 1 = x^2 - 1/3 => {1, x^2 - 1/3} orthogonal basis for W => proj_w (x^3) = / u1 + / u2