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Unformatted text preview: Practice Midterm 1 Problem 1. Let V = {(a1 , a2 ) : a1 , a2 ∈ I }. Deﬁne addition of elements R of V coordinatewise, and for (a1 , a2 ) in V and c ∈ I deﬁne R, c(a1 , a2 ) = (0, 0), if c = 0 (ca1 , ac2 ), if c = 0 Is V a vector space over I with these operations? Justify your answer. R 2 Problem 2. Determine whether the following set W = {(a1 , a2 , a3 ) ∈ R3 : 2a1 − 7a2 + a3 = 0} is a subspace of I 3 under the coordinatewise addition and scalar multipliR cation deﬁned on I 3 . R 3 Problem 3. Determine whether the vector (2, −1, 1) is the span of S = {(1, 0, 2), (−1, 1, 1)}. 4 Problem 4. Let u, v, and w be distinct vectors of a vector space V. Show that if {u, v, w} is linearly independent, then {u + v + w, v + w, w} is also linearly independent. 5 Problem 5. Verify if the following sets are linearly independent, generating sets, bases in I 3 . R (a) {(1, 0, −1), (2, 5, 1), (0, −4, 3)} (b) {(1, 2, −1), (1, 0, 2), (2, 1, 1)} 6 ...
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This note was uploaded on 01/28/2011 for the course MATH 115 taught by Professor Boisvert during the Spring '08 term at UCLA.
 Spring '08
 Boisvert
 Addition, Vector Space

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