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M415m2_ans - Midterm II MATH 415 B13 B14 Linear Algebra...

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Midterm II MATH 415, B13, B14 - Linear Algebra Fall 2007 Oct. 17, 2007 Name: This exam has 6 questions, for a total of 110 points. Please show ALL your work. Question: 1 2 3 4 5 6 Total Points: 15 15 20 25 25 10 110 Score: 1

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Math 415 Midterm II 1. (15 points) Determine whether the following vectors are linearly independent in R 2 × 2 . - 1 - 1 2 0 , 1 1 - 1 - 2 , 0 0 - 1 2 , 1 0 0 2 . Solution: No. Consider the determinant det - 1 - 1 2 0 1 1 - 1 - 2 0 0 - 1 2 1 0 0 2 = 0 So they are linearly dependent. 2. (15 points) Determine the kernel and range of the linear transformation L : P 3 P 4 defined by L ( p ( x )) = x 2 p ( x ) - xp ( x ) + p (0) Solution: Let p ( x ) = ax 2 + bx + c , then L ( p ( x )) = ax 3 - cx + c . Then for p ( x ) to be in the kernel, we have a = c = 0, so Kernel ( L ) = Span ( x ). The range of L is Span ( x 3 , - x + 1). Page 2 Please go on to the next page. . .
Math 415 Midterm II 3. (20 points) Let u 1 = 2 e 1 + 3 e 2 , u 2 = e 1 + 2 e 2 . Let v 1 = e 1 - e 2 , v 2 = e 1 - 2 e 2 . Then [ u 1 , u 2 ] and [ v 1 , v 2 ] are two ordered basis of R 2 . (a) Find the transition matrix corresponding the change of basis from [ u 1 , u 2 ] to [ v 1 , v 2 ].

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