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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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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.
 Winter '07
 Schilling
 Linear Algebra, Algebra, Vector Space

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