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Unformatted text preview: 1. Short Answer Problems [2] a) Let S = { v 1 , . . . , v n } be a nonempty subset of a vector space V . Define the statement S is linearly independent. Solution: S is linearly independent if the only solution of c 1 v 1 + c 2 v 2 + ··· + c n v n = 0 is c 1 = c 2 = ··· = c n = 0. [2] b) Write the definition of a basis for a vector space V . Solution: A set of vectors B = { v 1 , . . . , v n } is a basis for V if span( B ) = V and B is linearly independent. [1] c) Write the definition of the dimension of a vector space V . Solution: The dimension of V is the number of elements in any basis for V . [2] d) Prove that 0 x = for any x ∈ V . Solution: By vector space axioms 3,4,2,8 and 10, 0 x = 0 x + 0 = 0 x + ( x + ( x )) = (0 x + x ) + ( x ) = (0 + 1) x + ( x ) = 1 x + ( x ) = x + ( x ) = 0. [2] e) Is it true that if a set S with more than one vector is linearly dependent then every vector v ∈ S can be written as a linear combination of the other vectors. Justify your answer. Solution: It is false. Consider the set { (0 , 0) , (1 , 1) } . The set is linearly depenedent (since it contains the zero vector) and (1 , 1) can not be written as a linear combination of (0 , 0). Math 136  Term Test 1 Solutions Winter 2009 2. Let β = { x 2 4 x + 4 , x 2 , 1 } . [4] a) Show that span( β ) = P 2 . Solution: For any p, q, r ∈ R , if a ( x 2 4 x + 4) + b ( x 2) + c (1) = px 2 + qx + r , by comparing coefficients, we have a = p 4 a + b = q 4 a 2 b + c = r which implies that a = p , b = q + 4 p and c = r + 2 q + 4 p , and the system is consistent for any p, q, r . This shows that span β = P 2 . [2] b) Let w = x 2 + x + 1. Find the β coordinate vector of w . Solution: Let a ( x 2 4 x + 4) + b ( x 2) + c (1) = w = x 2 + x + 1 by a), a = 1, b = 5 and c = 7 therefore ( w ) β = (1 , 5 , 7) 2 Math 136  Term Test 1 Solutions Winter 2009 3. Determine, with proof, which of the following are subspaces of the given vector space....
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This note was uploaded on 01/30/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.
 Spring '08
 ANDREWCHILDS
 Algebra, Vector Space

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