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Math 110: Worksheet 1 Solutions

Math 110: Worksheet 1 SolutionsAugust 30 Thursday Aug. 241. Determine whether or not the following sets form vector spaces over the given fields.(a) The setVof all matrices of the form(1ab1)wherea, b R, overRwith standardaddition and scalar thatVis not closed under addition: fora, b, c, d R, we have(1ab1)and(1cd1)but(1ab1)+(1cd1)=(2a+ cb+d2)/ conclude thatVis not a vector space with the given operations.(b) The setVof all matrices of the form(1ab1)wherea, b R, overRwith addition and scalar multiplication defined by(1ab1) (1cd1)=(1a+cb+d1), k (1ab1)=(1kakb1).We claim thatVis indeed a vector space with the given operations. Note firstthatVis closed under the addtion and scalar multiplication operations: for(1ab1),(1cd1) Vandk R, we have(1ab1) (1cd1)=(1a+cb+d1) Vandk (1ab1)=(1kakb1) 1:Observe that(1ab1) (1cd1)=(1a+cb+d1)( defn. of addition)=(1c+ad+b1)( comm. of addition inR)=(1cd1) (1ab1)( addition)VS 2:Note that((1ab1) (1cd1)) (1ef1)=(1a+cb+d1) (1ef1)( defn.)

that V is closed under the addtion and scalar multiplication operations: for 1 a b 1 ; 1 c d 1 2V and k 2R, we have 1 a b 1 1 c d 1 = 1 a+ c b+ d 1 2V and k 1 a b 1 = 1 ka kb 1 2V: 1. VS 1: Observe that 1 a b 1 1 c ... product ee0. By thinking of e as an identity, we have ee0= e0. Likewise, thinking of

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