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**Unformatted text preview: **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 4 ,v 2 = 4 3 I ,v 3 = 1 . 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 ∈ C so that when λ = λ 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 6 = λ what condition on the...

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