Fall 2009 Midterm #2

# Fall 2009 Midterm #2 - Linear Algebra F09 Name Page 1 of...

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Linear Algebra F09 Name: Page 1 of 7 (1) (5 marks) Find parametric equations for the line passing the point P (1 , 1 , 2) and parallel to the two planes given below: 2 x y +3 x =0 ,x 3 y z 2=0 (2) (4 marks) Find the steady state vector q for the following regular tran- sition matrix: P = ± 1 / 41 / 5 3 / 44 / 5 ²

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Linear Algebra F09 Name: Page 2 of 7 (3) (4 marks) Say T : R 2 R 2 is the linear operator de±ned by: a reﬂection about the line y = x followed by a counter-clockwise rotation by an angle of π/ 2 followed by an othogonal projection onto the y -axis. Use [ T ]=[ T ( e 1 ) | T ( e 2 )] to get the standard matrix of T . (NOTE: for full marks, you must use this theorem.) (4) (3 marks) Let T : R 3 R 3 be the linear operator de±ned by T ( x 1 ,x 2 3 )=(2 x 1 + x 2 +2 x 3 , 2 x 1 x 2 +5 x 3 , 4 x 1 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 = { M 1 ,M 2 3 } is a basis for the vector space V of 2 × 2 symmetric matrices, where M 1 = ± 12 20 ² 2 = ± 0 1 10 ² 3 = ± 01 ²

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Fall 2009 Midterm #2 - Linear Algebra F09 Name Page 1 of...

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