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sample_final_2

# sample_final_2 - Math 136 Final Exam Spring 2009 NOTE The...

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Math 136 - Final Exam Spring 2009 NOTE: The questions on this exam does not exactly reflect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not asked here. 1. Short Answer Problems a) What can you say about the consistency and number of parameters in the general solution of a system of 4 linear equations in 5 variables if the rank of the coefficient matrix is 4. b) Let B = { x 2 + 1 , x 2 + 2 x, x + 1 } be a basis for P 2 . If [ ~v ] B = 1 2 3 , find ~v . c) Let ~x = (1 , - 1 , 2) and ~ y = ( - 2 , 1 , 1). Calculate ~x × ~ y and ~x · ~ y . d) List 3 things equivalent to an n × n matrix being invertible. e) Let ~a be any fixed vector in R n . Prove that proj ~a (perp ~a ( ~x )) = ~ 0 for all ~x R n . 2. Consider the system of linear equations x 1 + 2 x 2 + - x 4 = 2 - x 1 - 2 x 2 + x 3 + 4 x 4 = - 4 2 x 1 + 4 x 2 + 2 x 3 + 4 x 4 = 0 a) Write the coefficient matrix A and augmented matrix for the system.

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