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PracticeMidtermII

# PracticeMidtermII - Linear Algebra F09 Name Page 1 of 7(1(5...

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Linear Algebra F09 Name: Page 1 of 7 (1) (5 marks) Find all the unit vectors orthogonal to u 1 = (1 , - 2 , 1) and u 2 = (2 , 0 , - 1) . (2) (4 marks) Say T : R 2 R 2 is the linear operator defined by: a clockwise rotation by an angle of π/ 2 followed by a reflection about the line y = x followed by a dilation by a factor of 2. Use [ T ] = [ T ( e 1 ) | T ( e 2 )] to get the standard matrix of T . (NOTE: for full marks, you must use this theorem.)

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Linear Algebra F09 Name: Page 2 of 7 (3) (4 marks) Find the steady state vector q for the following regular tran- sition matrix: P = 2 / 5 1 / 4 3 / 5 3 / 4 (4) (3 marks) Let T : R 3 R 3 be the linear operator defined by T ( x 1 , x 2 , x 3 ) = (2 x 1 + x 2 + 2 x 3 , - 4 x 2 + 3 x 3 , 4 x 1 + 2 x 2 + 7 x 3 ) Determine if T is one-to-one.
Linear Algebra F09 Name: Page 3 of 7 (5) (5 marks) Determine whether the set S = { p 1 , p 2 , p 3 , p 4 } is a basis for the vector space P 4 of polynomials of degree 3 or less with real coefficients, where p 1 = 1 - x - x 2 , p 2 = 2 - x - 2 x 2 , p 3 = 3 - 3 x 2 , p 4 = 1 - x 2 + 2 x 3

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Linear Algebra F09 Name: Page 4 of 7 (6) (6 marks) Let V = M 22 , and let W
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PracticeMidtermII - Linear Algebra F09 Name Page 1 of 7(1(5...

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