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Transcript

Math102 Lab8 1. Determine whether the following sets are vector spaces under the given operations. For those that are not vector spaces list all axioms that fail to hold. a 1 (a) The set of all 2 × 2 matrices of the form with the standard matrix 1 b addition and scalar muptiplication. (b) The set of all pairs of real numbers of the form (1, x) with the operations (1, y) + (1, y 0 ) = (1, y + y 0 ) and k(1, y) = (1, ky). 2. Use theorem 5.2.1 to determine which of the following are subspaces of M22 (a) all 2 × 2 matrices with integer entries a b (b) all matrices where a + b + c + d = 0 c d 3. Express (−9, −7, −15) as linear combinations of u = (2, 1, 4), v = (1, −1, 3) and w = (3, 2, 5) 4. Is the set {(3, 8, 7, −3), (1, 5, 3, −1), (2, −1, 2, 6), (1, 4, 0, 3)} of vectors in R4 linearly independent? 5. Show that the vectors v1 = (0, 3, 1, −1), v2 = (6, 0, 5, 1) and v3 = (4, −7, 1, 3) form a linearly dependent set in R4 . 1 Solution 1. Look at the list of axioms on page 222. (a) Axioms 1,4,5 and 6 fail. (b) This set is a vector space with the zero element (1, 0) and the identity element as (1, 1). 2. Recall Theorem 5.2.1: If W is a set of one or more vectors from a vector space V , then W is a subspace of V if and only if the following conditions holds: (i) If u and v are vectors in W , then u + v is in W . (ii) If k is any scalar and u is any vector in W , then ku is in W . (a) If you take k as any real number then (ii) of the theorem fails, hence this set is not a subspace of M22 (b) Both (i) and (ii) holds for this set, hence it is a subspace of M22 3. (−9, −7, −15) = −2u + v − 2w 4. Yes it is linearly independent. 5. One can show that v3 = dependent set in R4 . 2 v 3 2 − 37 v1 , hence this shows they form a linearly 2