HW2 - MA/PH 607 Homework Handout II A For the following i...

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MA/PH 607 8/29/11 Homework Handout II A. For the following, is a three-dimensional space of traditional vectors with standard basis i f œ ß ß Þ e f i j k (If you prefer, use or .) e f e f e e e x y z 1 2 3 ß ß ß ß Also, let U œ ß ß e f b b b 1 2 3 where b i j b j k b i j " œ ° # ß œ $ ± œ # ± $ 2 3 and . 1. Solve for , and in terms of , and . i j k b b b 1 2 3 2. Is a basis for ? Give a reason for your answer. U i 3. Let . What is in terms of ? v i j k v œ # ° $ ° % U 4. Find the following (with as above): v l ß l ß l ß l ß l ß l ß l ß l ß l ß l ß l ß l ß l l i j k b b b v i j k b b b v ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ f f f f f f f U U U U U U U " " 2 3 2 3 and 5. Compute (i.e., ) for all possible 's and 's.   ¡ b b b b 3 4 3 4 l 3 4 6. Let and v b b b w b b b . œ @ ° @ ° @ œ A ° A ° A " " # $ " " # $ 2 3 2 3 Find the corresponding component formulas for and .   ¡ l l v w v l Œ   ¡ l l ° ± ° ± ° ± É Note: and ! v w v l Á @ A ° @ A ° @ A Á @ ° @ ° @ " " # # $ " # $ 3 2 2 2 7. (optional) Suppose is any vector in and let the components of with respect to c c i our two bases be denoted by l œ l œ c c ¡ ¡ Ô × Ô × Õ Ø Õ Ø f U α " α " α " " " # # $ $ and .
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