Math416_Midterm1_Practice_sol - Math 416 - Abstract Linear...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Practice midterm 1 Name: Solutions • This is a practice exam. The real exam will consist of at most 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. /10 2. /10 3. /10 4. /10 5. /10 6. /10 7. /10 8. /10 Total: /80 1 Problem 1a. (5 pts) Find all solutions (if any) of the system x 1 + 3 x 2 + 2 x 3 = 2 x 1 + 6 x 2 + x 3 = 3 2 x 1 + 3 x 2 + 5 x 3 = 5 . 1 3 2 2 1 6 1 3 2 3 5 5 ∼ 1 3 2 2 3- 1 1- 3 1 1 ∼ 1 3 2 2 0 3- 1 1 0 0 2 Because of the last row, there is no solution. b. (5 pts) Let A = 1 3 2 1 6 1 2 3 5 . Find a basis of Null A . 1 3 2 1 6 1 2 3 5 ∼ 1 3 2 0 3- 1 0 0 ∼ 1 0 3 0 3- 1 0 0 Null A = { - 3 x 3 1 3 x 3 x 3 | x 3 ∈ R } has basis { - 9 1 3 } . Remark: The basis { - 3 1 3 1 } is just as good! 2 Problem 2. Let C ( R ) denote the vector space of all continuous functions f : R → R . Let T : C ( R ) → R be the transformation defined by T ( f ) = integraldisplay 2 1 f ( x ) dx . a. (4 pts) Is T linear? Prove your answer. Yes it is. Take f, g ∈ C ( R ) and scalars α, β ∈ R . Then we have T ( αf + βg ) = integraldisplay 2 1 ( αf + βg )( x ) dx = integraldisplay 2 1 αf ( x ) + βg ( x ) dx = α integraldisplay 2 1 f ( x ) dx + β integraldisplay 2 1 g ( x ) dx = αT ( f ) + βT ( g ) . b. (4 pts) Consider the subspace P 2 ⊂ C ( R ) of polynomial functions of degree at most 2. Find the matrix representing T : P 2 → R with respect to the monomial basis { 1 , x, x 2 } of P 2 . T (1) = integraldisplay 2 1 1 dx = [ x ] 2 1 = 2- 1 = 1 T ( x ) = integraldisplay 2 1 x dx = bracketleftbigg 1 2 x 2 bracketrightbigg 2 1 = 1 2 (4- 1) = 3 2 T ( x ) = integraldisplay 2 1 x 2 dx = bracketleftbigg 1 3 x 3 bracketrightbigg 2 1 = 1 3 (8- 1) = 7 3 The matrix representing...
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This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.

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Math416_Midterm1_Practice_sol - Math 416 - Abstract Linear...

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