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LAHW#13

LAHW#13. Due January 2, 2012. 7.1. Inner-Product Spaces. 3.

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LAHW#13

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  1. LAHW#13 Due January 2, 2012

  2. 7.1. Inner-Product Spaces • 3. • Show that the vectors u = (2, 3, -5), v = (2, 2, 2), and w = (-8, 7, 1) form an orthogonal triple; that is, each vector is orthogonal to the other two. Find the projection of x = (4, -3, 5) onto each of the vectors u, v, and w. Verify that the three projections add up to give x. Thus the vector x will be dissected into three mutually orthogonal pieces or components.

  3. 7.1. Inner-Product Spaces • 6. • Let x = (1, 3, 7) and y = (-4, 2, 1). Compute ||x||, <x, y>, ||y||, and ||x + y||. Verify in this example that the Cauchy-Schwarz inequality and the triangle inequality are true.

  4. 7.1. Inner-Product Spaces • 10. • Consider the matrixfind s simple description of the null space and the orthogonal complement of the row space.

  5. 7.1. Inner-Product Spaces • 15. • In an inner-product space, if <x, y> = 0,then ||x + y||2 = ||x||2 + ||y||2. Establish this and determine whether the converse is also true.

  6. 7.1. Inner-Product Spaces • 24. • In any inner-product space, if ||x||2 = ||y||2 = <x, y>, then x = y. Explain why or why not.

  7. 7.1. Inner-Product Spaces 31. Let A be a square matrix whose columns form an orthogonal set. If we normalize each row of A, we obtain a new matrix, B. Is B an orthogonal matrix? What interesting properties does B have? Answer the same questions for the matrix C obtained by normalizing the columns of A.

  8. 7.1. Inner-Product Spaces • 42. • Establish this identity in any real inner-product space: ||x – y||2 = ||x||2 – ||y||2 + 2<y – x, y>.

  9. 7.1. Inner-Product Spaces • 60.

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