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Math416_Midterm2_sol - Math 416 Abstract Linear Algebra...

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Math 416 - Abstract Linear Algebra Fall 2011, section E1 Midterm 2, October 19 Name: Solutions No calculators, electronic devices, books, or notes may be used. Show your work. No credit for answers without justification. Good luck! 1. /10 2. /10 3. /8 4. /12 Total: /40 1
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Problem 1. Let T : R 2 R 2 be a linear map and { v 1 , v 2 } a basis of R 2 satisfying Tv 1 = 4 v 1 - 3 v 2 Tv 2 = v 1 + 2 v 2 . a. (1 pt) Compute T (10 v 1 + 3 v 2 ). T (10 v 1 + 3 v 2 ) = 10 Tv 1 + 3 Tv 2 = 10(4 v 1 - 3 v 2 ) + 3( v 1 + 2 v 2 ) = 40 v 1 - 30 v 2 + 3 v 1 + 6 v 2 = 43 v 1 - 24 v 2 . b. (1 pt) Write the matrix representing T in the basis { v 1 , v 2 } . (No justification needed.) [ T ] { v i } = 4 1 - 3 2 2
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c. (6 pts) Given v 1 = 2 3 and v 2 = 1 3 , find the standard matrix representation of T . By the change of basis formula and (b), we have [ T ] standard = V [ T ] { v i } V - 1 = 2 1 3 3 4 1 - 3 2 2 1 3 3 - 1 = 5 4 3 9 1 3 3 - 1 - 3 2 = 1 3 3 3 - 18 15 = 1 1 - 6 5 . d. (2 pts) Compute T ( e 1 - e 2 ). T ( e 1 - e 2 ) = 1 - 6 - 1 5 = 0 - 11 . 3
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Problem 2. (10 pts) Let B be an n × p matrix whose columns span R n , and let A be any m × n matrix. Prove rank AB = rank A .
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