PS2 - Mathematic 104 Fall 2010 Assignment#2 Due Wednesday...

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Mathematic 104, Fall 2010: Assignment #2 Due: Wednesday, October 13th 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. Excercise 1.3 of Lecture 1 of Trefethen-Bau. Problem #2. Consider the following three vectors in C 3 : v 1 = 3 I 0 4 , v 2 = 4 0 3 I , v 3 = 0 1 0 . a) Show that { v 1 , v 2 , v 3 } is an orthogonal set. Is this set orthonormal? b) Let X C 3 × 3 be a matrix so that Xv 1 = v 2 + v 3 , Xv 2 = - v 2 and Xv 3 = v 1 + v 2 + v 3 . Determine X . (Hint: Look for a natural orthonormal basis). Problem #3. Let v 1 , v 2 and v 3 be vectors in C 3 . Determine a value λ 0 C so that when λ = λ 0 the vectors w 1 = v 1 + v 2 , w 2 = v 1 - v 3 and w 3 = λv 1 + v 2 + v 3 are never a basis of C 3 . When λ = λ 0 what condition on the { v i } is neccesary and sufficient so that the { w i } form a basis?
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