exam_sol - Winter 2011 Math 67 Linear Algebra Midterm Exam...

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Unformatted text preview: Winter 2011 Math 67 Linear Algebra Midterm Exam February 9, 2011 Instructions. You have 50 minutes to complete the 5 problems on the exam. All of your answers should be written in complete English sentences. The symbol V will always denote a vector space over the real numbers R . No books, notes, calculators, talking, texting, etc. is allowed. Problem 1. (a) (3 points) Complete the definition: The vectors ( v 1 ,...,v n ) are linearly independent if . .. (b) (2 points) Complete the definition: The vectors ( v 1 ,...,v n ) form a basis of V if . .. (c) (5 points) Let ( v 1 ,v 2 ,v 3 ) be a basis of V . Prove that ( v 1 ,v 2 ,v 1 + v 3 ) is also a basis of V . (a) ... a 1 v 1 + a 2 v 2 + + a n v n = 0 implies that a 1 = a 2 = = a n = 0. (b) ...they span V and are linearly independent. (c) Proof. Since v 3 = ( v 1 + v 3 )- v 1 , we have that span { v 1 ,v 2 ,v 1 + v 3 } = span { v 1 ,v 2 ,v 3 ,v 1 + v 3 } = span { v 1 ,v 2 ,v 3 } = V....
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exam_sol - Winter 2011 Math 67 Linear Algebra Midterm Exam...

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