PS5 - Mathematic 104 Fall 2010 Assignment#5 Due Wednesday...

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Mathematic 104, Fall 2010: Assignment #5 Due: Wednesday, November 10th Instructions: Please ensure that your answers are legible. Also make sure that all steps are shown – even for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good practice, are not required and will not be graded. Problem #1. Let A R 2 × 2 be the following matrix A = 1 2 0 - 1 Denote by S p = { u R 2 : || u || p = 1 } the set of vectors of length 1 in the p -norm and let AS p = { Au R 2 : u S p } be the image under A of S p . Here 1 p ≤ ∞ . a) Determine || A || 1 and find vectors u S 1 and v = Au AS 1 so that || v || 1 = || Au || 1 = || A || 1 . Sketch S 1 and AS 1 and indicate the vectors u and v on the sketch. b) Determine || A || and find vectors u S and v = Au AS so that || v || = || Au || = || A || . Sketch S and AS and indicate the vectors u and v on the sketch. c) Determine || A || 2 and find vectors u S 2 and v = Au AS 2 so that || v || 2 = || Au || 2 = || A || 2 . Sketch S 2 and AS 2 and indicate the vectors u and v on the sketch. To do this it is useful to use that any vector u S 2 may be written as u = cos θ sin θ . Problem #2. Suppose that P C m × m and P 2 = P (so P is an oblique projector). Show that as long as
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