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Fall 2005 - Wickramasekera's Class - Practice Exam 1

Fall 2005 - Wickramasekera's Class - Practice Exam 1 - MATH...

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MATH 20F: LINEAR ALGEBRA, WINTER 2007 MIDTERM 1 PRACTICE PROBLEMS Note: This is not a practice exam; you don’t have to be able to do these problems all in 50 minutes. Some of these problems are longer than any of the problems on the actual exam. (1) Find the solution(s) to the following system of linear equations by row reducing the associated augmented matrix. If the system has more than one solution, express the set of solutions in parametric vector form. x 1 - x 2 + 2 x 3 - x 4 = - 1 2 x 1 + x 2 - 2 x 3 - 2 x 4 = - 2 - x 1 + 2 x 2 - 4 x 3 + x 4 = 1 3 x 1 - 3 x 4 = - 3 (2) For which value(s) of a does the following system have zero, one, in- finitely many solutions? x 1 + 2 x 2 - 3 x 3 = 4 3 x 1 - x 2 + 5 x 3 = 2 4 x 1 + x 2 + ( a 2 - 14) x 3 = a + 2 (3) Determine whether the following set of vectors in R 3 is linearly inde- pendent. { (2 , - 1 , 4) , (3 , 6 , 2) , (2 , 10 , - 4) } . (Note that here we have used the al- ternate notation of writing vectors as rows rather than as columns.) 1
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(4) Let u , v and w be any three vectors in R n . Prove that the vectors u - v , v - w and w - u form a linearly dependent set.
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