MTH 309Y 37. Inner product spaces. = a 1 b 1 + a 2 b a n b n

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1 MTH 39Y 37. Inner product spaces Recall: ) The dot product in R n : a. a n b. b n = a b + a 2 b a n b n 2) Properties of the dot product: a) u v = v u b) (u + v) w = u w + v w c) (cu) v = c(u v) d) u u and u u = if and only if u =. 2) Using the dot product we can define: length of vectors distance between vectors orthogonality of vectors orthogonal and orthonormal bases orthogonal projection of a vector onto a subspace of R n... Next: Generalization to arbitrary vector spaces. 232

2 Definition Let V be a vector space. An inner product on V is a function V V R u, v u, v such that: a) u, v = v, u b) u + v, w = u, w + v, w c) cu, v = c u, v d) u, u and u, u = if and only if u =. Definition Let V be a vector space with an inner product,. ) The length (or norm) of a vector v is the number v = v, v 2) The distance between vectors u, v V is the number dist(u, v) = u v 3) Vectors u, v V are orthogonal if u, v =. 233

3 Example. The dot product is an inner product in R n. Example. Let p,..., p n be any positive numbers. For vectors u, v R n u = a. a n v = b. b n define: u, v = p (a b ) + p 2 (a 2, b 2 ) p n (a n b n ) This gives an inner product in R n. 234

4 Example. Let C[, ] be the vector space of continuous functions f : [, ] R. Define: f, g = This is an inner product on C[, ]. f(t)g(t)dt y = g(t) y = f(t) Example. Compute the length of the function f(t) = + t 2 in C[, ]. 235

5 Definition Let V be a vector space with an inner product,, and let W be a subspace of V. A vector v V is orthogonal to W if v, w = for all w W. Definition Let V be a vector space with an inner product,, and let W be a subspace of V. The orthogonal projection of a vector v V onto W is a vector proj W v such that ) proj W v W 2) the vector z = v proj W v is orthogonal to W. Best Approximation Theorem If V is a vector space with an inner product,, W is a subspace of V, and v V, then proj W v is the vector of V which is the closest to v: for all w W. dist(v, proj W v) dist(v, w) Theorem Let V is a vector space with an inner product,, and let W be a subspace of V. If B = {w,..., w k } is an orthogonal basis of W (i.e. a basis such that w i, w j = for all i j) then for v V we have: proj W v = v, w w, w w v, w k w k, w k w k 236

6 Application: Fourier approximations. Goal: Let f : [, ] R be a continuous function. approximation of f of the form Find the best possible P(t) = a + a sin(2πt) + b cos(2πt) + a 2 sin(2π2t) + b 2 cos(2π2t) a n sin(2πnt) + b n cos(2πnt) sin(2πt) sin(2π2t) sin(2π3t) cos(2πt) cos(2π2t) cos(2π3t) Note: Let W n be a subspace of C[, ] given by: W n = Span(, sin(2πt), cos(2πt),..., sin(2πnt), cos(2πnt)) By the Best Approximation Theorem, the best approximation of f is obtained if we take P(t) = proj Wn f(t). 237

7 Theorem The set {, sin(2πt), cos(2πt),..., sin(2πnt), cos(2πnt)} is an orthogonal basis of W n. Corollary If f C[, ] then where: a = f,, = f(t)dt and for k > : a k = b k = proj Wn f(t) = a + a sin(2πt) + b cos(2πt) + a 2 sin(2π2t) + b 2 cos(2π2t) a n sin(2πnt) + b n cos(2πnt) f, sin(2πkt) sin(2πkt), sin(2πkt) = 2 f(t) sin(2πkt)dt f, cos(2πkt) cos(2πkt), cos(2πkt) = 2 f(t) cos(2πkt)dt 238

8 Example. Compute proj Wn f(t) for the function f(t) = t. 239

9 Application: Polynomial approximations. Goal: Let f : [, ] R be a continuous function. Find the best possible approximation of f given by a polynomial P(t) of degree n: P(t) = a + a t a n t n Note: Let P n be the subspace of C[, ] consisting of all polynomials of degree n: P n = {a + a t a n t n a k R} By the Best Approximation Theorem, the best approximation of f is obtained if we take P(t) = proj Pn f(t). 24

10 Gram-Schmidt process: a basis {v,..., v k } of W V G-S process an orthogonal basis {w,..., w k } of W Theorem (Gram-Schmidt Process) Let V be a vector space with an inner product,, and let W be a subspace of V. Let {v,..., v k } be a basis of W. Define vectors {w,..., w k } as follows: w = v w 2 = v 2 w, v 2 w, w w w 3 = v 3 w, v 3 w, w w w 2, v 3 w 2, w 2 w 2 w k = v k w, v k w, w w w 2, v k w 2, w 2 w 2... w k, v k w k, w k w k Then the set {w,..., w k } is an orthogonal basis of W. 24

11 Example. C[, ]. Find an orthogonal basis of the subspace P 2 of the vector space 242

12 Example. Compute proj P2 f(t) for f(t) = t. 243