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# solution3 - subspaces of Rn(n 3(a all such that a1 0 pp...

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Homework assignment 3 pp. 39-40 Exercise 1. Which of the following sets S of vectors α = ( a 1 ,...,a n ) R n are subspaces of R n ( n 3 )? (a) all α such that a 1 0; No. Take α = (1 , 0 ,..., 0) S , then ( - α ) = ( - 1 , 0 ,...,n ) / S. (b) all α such that a 1 + 3 a 2 = a 3 ; Yes. If α = ( a 1 ,...,a n ) = ( b 1 ,...,b n ) S , then α + β,λα S , since ( a 1 + b 1 ) + 3( a 2 + b 2 ) = = ( a 1 + 3 a 2 ) + ( b 1 + 3 b 2 ) = a 3 + b 3 and λa 1 + 3 λa 2 = λ ( a 1 + 3 a 3 ) = λa 2 . (c) all α such that a 2 = a 2 1 ; No. Take α = (1 , 1 , 0 ,..., 0) S , then 2 α = (2 , 2 , 0 ,..., 0) / S . (d) all α such that a 1 a 2 = 0; No. Take α = (0 , 1 , 0 ,..., 0) = (1 , 0 ,..., 0) S , then α + β = (1 , 1 , 0 ,..., 0) / S . (e) all α such that a 2 is rational. No. Take α = (0 , 1 , 0 ,..., 0) , then 2 α / S . Exercise 2. Let V be the (real) vector space of all functions f from R into R . Which of the following sets of functions are subspaces of V ? (a) all f such that f ( x 2 ) = f ( x ) 2 ; No. Take a constant function f ( x ) = 1 for all x . Then f S , but 2 f S . (b) all f such that f (0) = f (1); Yes. If f,g S , then f + g,λg S , since ( f + g )(0) = f (0) + g (0) = f (1) + g (1) = ( f + g )(1) , and λf (0) = λf (1) . (c) all f such that No. Take a function f such that f (3) = 1 and f (3) = 1 + f ( - 5); f ( x ) = 0 for all x 6 = 3 . Then f S , but 2 f / S . (d) all

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solution3 - subspaces of Rn(n 3(a all such that a1 0 pp...

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