aem570-2010-hw3 - Q ( v ) = 2 Q ( v ) for each R and v V ....

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AA/EE/ME 570: Manifolds and Geometry for Systems and Control Homework #3 1. Show that a norm k · k on a vector space V is the natural norm associated to an inner product if and only if k · k satisfies the parallelogram law : k u + v k 2 + k u - v k 2 = 2( | u k 2 + k v k 2 ) . 2. Consider R 3 with the standard inner product and let U R 3 be a two-dimensional subspace. (a) Define explicitly the orthogonal projection P U onto U . (b) Define explicitly the reflection map R U that reflects vectors about U . 3. Let A,B R n × n and define << A,B >> = tr ( AB T ). (a) Show that << · , · >> is an inner product on R n × n . (b) Show that the subspaces of symmetric and skew-symmetric matrices are orthogonal with respect to this inner product. (c) Show that the orthogonal projection sym : R n × n R n × n onto the set of symmetric matrices is given by sym ( A ) = 1 2 ( A + A T ). (d) Show that the orthogonal projection skew : R n × n R n × n onto the set of skew-symmetric matrices is given by skew ( A ) = 1 2 ( A - A T ). 4. Let V be an R -vector space, and let B Σ 2 ( V ). (a) Prove the polarization identity: 4 B ( v 1 ,v 2 ) = B ( v 1 + v 2 ,v 1 + v 2 ) - B ( v 1 - v 2 ,v 1 - v 2 ) . A quadratic function on V is a function Q : V R satisfying
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Unformatted text preview: Q ( v ) = 2 Q ( v ) for each R and v V . (b) Show how the polarization identity provides a means of dening a symmetric bilinear map given a quadratic function on a vector space. 5. Let [ a,b ] be an interval and let L 2 ([ a,b ]) be the set of functions f : [ a,b ] R whose square is integrable over [ a,b ]. We refer to such functions as square-integrable . Show that the map ( f,g ) 7 R b a f ( t ) g ( t ) dt is an inner product. This map is referred to as the L 2 ([ a,b ])-inner product; it is convenient to denote it by f,g L 2 ([ a,b ]) . 6. For V a nite-dimensional R-vector space, show that the following pairs of vector spaces are naturally isomorphic by providing explicit isomorphisms: (a) T 1 ( V ) and V ; (b) T 1 ( V ) and V * ; (c) T 1 1 ( V ) and L ( V ; V ); (d) T 2 ( V ) and L ( V ; V * ); (e) T 2 ( V ) and L ( V * ; V ). 1...
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