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3.1 - Math 2940 Solutions Fall 2011 Section 3.1 2 Since the...

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Math 2940 Solutions, Fall 2011 Section 3 . 1 2 ) Since the addition is usual, the first four conditions are satisifed as they don’t involve the scalar multiplication. (5) fails as 1 · (3 , 8) = (3 , 0) 6 = (3 , 8). (6) holds as ( c 1 c 2 )( x 1 , x 2 ) = ( c 1 c 2 x 1 , 0) = c 1 ( c 2 x 1 , 0) = c 1 ( c 2 ( x 1 , x 2 )) . (7) holds as c (( x 1 , x 2 ) + ( y 1 + y 2 )) = c ( x 1 + y 1 , x 2 + y 2 ) = ( cx 1 + cy 1 , 0) = ( cx 1 , 0) + ( cy 1 , 0) = c ( x 1 , x 2 ) + c ( y 1 , y 2 ). (8) holds as ( c 1 + c 2 )( x 1 , x 2 ) = (( c 1 + c 2 ) x 1 , 0) = ( c 1 x 1 + c 2 x 1 , 0) = ( c 1 x 1 , 0) + ( c 2 , x 1 , 0) = c 1 ( x 1 , x 2 ) + c 2 ( x 1 , x 2 ). 4 ) The ‘zero’ vector in the space of 2 × 2 matrices is ± 0 0 0 0 ² . 1 2 A = ± 1 - 1 1 - 1 ² and - A = ± - 2 2 - 2 2 ² . The smallest subspace containing A consists of all scalar multiples of A , namely all matrices of the from ± 2 r - 2 r 2 r - 2 r ² as r runs through all real numbers. 10 ) (a) Subspace (b) Not a subspace - does not contain ~ 0. (c) Not a subspace - (0 , 1 , 1) , (1 , 0 , 0) are in the set, but their sum is not. (d) Subspace (e) Subspace (f) Not a subspace - (1 , 2 , 3) is in the set, but it’s scalar multiple - 1(1 , 2 , 3) = ( - 1 , - 2 , - 3) is not in the set.
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