Mathematics_1_oneside.pdf

P roof let u u 1 u n 1 3 u u i j u i u j δ i j i i j

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P ROOF . Let U = ( u 1 ,..., u n ). (1) (3) [ U 0 U ] i j = u 0 i u j = δ i j = [ I ] i j , i.e., U 0 U = I . By Theorem 6.16 , U 0 U = UU 0 = I and thus U - 1 = U 0 . (3) (4) k Ux k 2 = ( Ux ) 0 ( Ux ) = x 0 U 0 Ux = x 0 x = k x k 2 . (4) (1) Let x , y R n . Then by (4), k U ( x - y ) k = k x - y k , or equivalently x 0 U 0 Ux - x 0 U 0 Uy - y 0 U 0 Ux + y 0 U 0 Uy = x 0 x - x 0 y - y 0 x + y 0 y . If we again apply (4) we can cancel out some terms on both side of this equation and obtain - x 0 U 0 Uy - y 0 U 0 Ux = - x 0 y - y 0 x . Notice that x 0 y = y 0 x by Theorem 8.2 . Similarly, x 0 U 0 Uy = ( x 0 U 0 Uy ) 0 = y 0 U 0 Ux , where the first equality holds as these are 1 × 1 matrices. The second equality follows from the properties of matrix multiplication (The- orem 4.15 ). Thus x 0 U 0 Uy = x 0 y = x 0 Iy for all x , y R n . Recall that e 0 i U 0 = u 0 i and Ue j = u j . Thus if we set x = e i and y = e j we obtain u 0 i u j = e 0 i U 0 Ue j = e 0 i e j = δ i j that is, the columns of U for an orthonormal system. (2) (3) Can be shown analogously to (1) (3). (3) (2) Let v 0 1 ,..., v 0 n denote the rows of U . Then v 0 i v j = [ UU 0 ] i j = [ I ] i j = δ i j i.e., the rows of U form an orthonormal system. This completes the proof.
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S UMMARY 61 — Summary • An inner product is a bilinear symmetric positive definite function V × V R . It can be seen as a measure for the angle between two vectors. • Two vectors x and y are orthogonal (perpendicular, normal) to each other, if their inner product is 0. • A norm is a positive definite, positive scalable function V [0, ) that satisfies the triangle inequality k x + y k ≤ k x k+k y k . It can be seen as the length of a vector. • Every inner product induces a norm: k x k = p x 0 x . Then the Cauchy-Schwarz inequality | x 0 y | ≤ k x k · k y k holds for all x , y V . If in addition x , y V are orthogonal, then the Pythagorean theo- rem k x + y k 2 = k x k 2 +k y k 2 holds. • A metric is a bilinear symmetric positive definite function V × V [0, ) that satisfies the triangle inequality. It measures the dis- tance between two vectors. • Every norm induces a metric. • A metric that is induced by an inner product is called an Euclidean metric . • Set of vectors that are mutually orthogonal and have norm 1 is called an orthonormal system . • An orthogonal matrix is whose columns form an orthonormal sys- tem. Orthogonal maps preserve angles and norms.
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P ROBLEMS 62 — Problems 8.1 Prove Theorem 8.2 . 8.2 Complete the proof of Theorem 8.6 . That is, show that equality holds if and only if x and y are linearly dependent. 8.3 (a) The Minkowski inequality is also called triangle inequal- ity . Draw a picture that illustrates this inequality. (b) Prove Theorem 8.7 . (c) Give conditions where equality holds for the Minkowski in- equality. H INT : Compute k x + y k 2 and apply the Cauchy-Schwarz inequality. 8.4 Show that for any x , y R n fl fl fl k x k-k y k fl fl fl ≤ k x - y k . H INT : Use the simple observation that x = ( x - y ) + y and y = ( y - x ) + x and apply the Minkowski inequality.
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