Math416_Midterm2_Practice - Math 416 - Abstract Linear...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Practice midterm 2 Name: This is a (long) practice exam. The real exam will consist of 4 problems. In the real exam, no calculators, electronic devices, books, or notes may be used. Show your work. No credit for answers without justification. Good luck! 1. /15 2. /10 3. /10 4. /10 5. /5 6. /10 7. /15 8. /10 9. /10 10. /10 11. /10 12. /5 Total: /120 1 Section 2.5 Problem 1. Let A be an m n matrix. a. (5 pts) Show that A has linearly independent columns if and only if A : R n R m preserves linear independence, in the following sense: For any collection of vectors v 1 , . . ., v k R n we have { v 1 , . . ., v k } is linearly independent { Av 1 , . . ., Av k } is linearly independent. 2 b. (5 pts) Show that A : R n R m preserves linear independence if and only if for every subspace S R n we have dim AS = dim S ....
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Math416_Midterm2_Practice - Math 416 - Abstract Linear...

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