1.2-1.4a

# 1.2-1.4a - 1 Math 110 Homework 1 Partial Solutions If you...

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1 Math 110 Homework 1 Partial Solutions If you have any questions about these solutions, or about any problem not solved, please ask via email or in oﬃce hours, etc. 1.2.13 This is not a vector space; it fails (at least) VS4. The zero element here is the pair (0,1). With the “addition” deﬁned here, the pair (1,0) has no additive inverse. In other words, there is no ( a 1 , a 2 ) such that ( a 1 , a 2 ) + (1 , 0) = (0 , 1). 1.2.21 This is very straightforward; I will sketch the details. The zero of Z is the pair (0 V , 0 W ). All of the axioms can be veriﬁed by applying the analogous axioms for V and W . As a cultural aside, this object Z is called the direct sum of V and W , denoted Z = V W . 1.3.10 W 1 can be checked to be a subspace. First, 0 = (0 , 0 , . . . , 0) W 1 since 0 + 0 + ··· + 0 = 0. Second, if x = ( a 1 , . . . , a n ) and y = ( b 1 , . . . , b n ) are in W 1 , then x + y = ( a 1 + b 1 , . . . , a n + b n ) W 1 since ( a 1 + b 1 ) + ··· + ( a n + b n

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1.2-1.4a - 1 Math 110 Homework 1 Partial Solutions If you...

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