2016 Pearson Education Inc Orthogonality and Least Squares THE GRAMSCHMIDT PROCESS Slide 67 2 2016 Pearson Education Inc INNER PRODUCT SPACES Definition An inner product on a vector space ID: 769354
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© 2016 Pearson Education, Inc. Orthogonality and Least Squares THE GRAM-SCHMIDT PROCESS
Slide 6.7- 2 © 2016 Pearson Education, Inc. INNER PRODUCT SPACES Definition An inner product on a vector space V is a function that, to each pair of vectors u and v in V, associates a real number and satisfies the following axioms, for all u, v, w in V and all scalars c: + and if and only u = 0A vector space with an inner product is called an inner product space.
Slide 6.7- 3 © 2016 Pearson Education, Inc. INNER PRODUCT SPACES Example 1 Fix any two positive numbers—say, 4 and 5—and for vectors u = (u1, u2) and v = (v1, v2) in , setShow that equation (1) defines an inner product.Solution Certain Axiom 1 is satisfied, because =. (1)
Slide 6.7- 4 © 2016 Pearson Education, Inc. INNER PRODUCT SPACES If w = (w1, w2), then ) w2w2+ w2 = + This verifies Axiom 2. For Axiom 3, compute c ( ) =
Slide 6.7- 5 © 2016 Pearson Education, Inc. INNER PRODUCT SPACES For Axiom 4, note that , and only if u1 = u2 = 0, that is, if u = 0. Also, So (1) defines an inner product on.
Slide 6.7- 6 © 2016 Pearson Education, Inc. LENGTHS, DISTANCES, AND ORTHOGONALITY Let V be an inner product space, with the inner product denoted by . Just as in, we define the length, or norm, of a vector v to be the scalarEquivalently, .A unit vector is one whose length is 1. The distance between u and v is . Vectors u and v are orthogonal if = 0.
Slide 6.7- 7 © 2016 Pearson Education, Inc. TWO INEQUALITIES Given a vector v in an inner product space V and given a finite-dimensional subspace W, we may apply the Pythagorean Theorem to the orthogonal decomposition of v with respect to W and obtain + See Fig 2 on the next slide. In particular, this shows that the norm of the projection of v onto W does not exceed the norm of v itself. This simple observation leads to the following important inequality.
Slide 6.7- 8 © 2016 Pearson Education, Inc. TWO INEQUALITIES Theorem 16 The Cauchy-Schwarz Inequality: For all u, v in V, (4)
Slide 6.7- 9 © 2016 Pearson Education, Inc. TWO INEQUALITIES Proof If u = 0, then both sides of (4) are zero, and hence the inequality is true in this case.If u 0, let W be the subspace spanned by u.Recall that |c| for any scalar c. Thus = =Since , we have , which gives (4).
Slide 6.7- 10 © 2016 Pearson Education, Inc. TWO INEQUALITIES Theorem 17 The Triangle Inequality: For all u, v in V,Proof2 = 2 + 2|| + 2 2 + 2+ 2 = ( ) 2 The triangle inequality follows immediately by taking square roots of both sides.