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201s02fin[1]

# 201s02fin[1] - B U Department of Mathematics Math 201...

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B U Department of Mathematics Math 201 Matrix Theory Spring 2002 Final Exam This archive is a property of Bo˘ gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Consider in R 3 the vectors: u = 1 1 0 , v = 2 2 1 , w = 0 0 2 , and b = 3 4 5 . (a) Determine whether { u, v, w } is a linearly independent set. (b) Determine whether b is in the subspace spanned by { u, v, w } . Solution: (a) Clearly not linearly independent, since v = 2 u + 1 2 w . (b) Consider A . . . b = 1 2 0 . . . 3 1 2 0 . . . 4 0 1 2 . . . 5 1 2 0 . . . 3 0 0 0 . . . 1 0 1 2 . . . 5 1 2 0 . . . 3 0 1 2 . . . 5 0 0 0 . . . 1 so Rank ( A . . . b ) = 3 and Rank ( A ) = 2. b is not in the Columnspace ( A ). Therefore, b is not in the subspace of R 3 which is spanned by { u, v, w } .

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