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Unformatted text preview: ASSIGNMENT 7 Solutions Let V be a vector space, and V 1 , V 2 ⊆ V be subspaces. The sum of V 1 and V 2 is the vector space V 1 + V 2 = { v 1 + v 2 : v 1 ∈ V 1 and v 2 ∈ V 2 } . 1. [4 marks] Prove that if V 1 ∩ V 2 = { } , (1) then dim ( V 1 + V 2 ) = dim V 1 + dim V 2 . (2) (You do not need to show that V 1 + V 2 is a vector space.) Note that if either V 1 or V 2 is the trivial subspace, the result is automatic. Suppose neither is the trivial subspace. Let B 1 = { x 1 , . . . , x m } and B 2 = { y 1 , . . . , y n } be bases for V 1 and V 2 , respectively. We claim that B = { x 1 , . . . , x m , y 1 , . . . , y n } is a basis for V 1 + V 2 . It certainly spans V 1 + V 2 ; it remains to show that it is linearly independent. Set = a 1 x 1 + ··· + a m x m + b 1 y 1 + ··· + b n y n . (3) For convenience, let x = a 1 x 1 + ··· + a m x m and y = b 1 y 1 + ··· + b n y n . We have three cases. Suppose x = . Since B 1 is linearly independent, the coefficients a 1 , . . . , a m are all equal to zero. Moreover, (3) may be rewritten as y = ; and since B 2 is linearly independent, the coefficients b 1 , . . . , b n are all equal to zero....
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This note was uploaded on 04/07/2010 for the course MATH 136 taught by Professor All during the Fall '08 term at Waterloo.
 Fall '08
 All
 Vector Space

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