assign6_soln

assign6_soln - Math 136 Assignment 6 Solutions 1....

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Unformatted text preview: Math 136 Assignment 6 Solutions 1. Determine, with proof, which of the following are subspaces of the given vector space. Find a basis for each subspace. a) S 1 = x 1 x 2 x 3 R 3 | x 1 + x 2 = 0 ,x 3 =- x 2 of R 3 . Solution: We have 0 + 0 = 0 and 0 =- 0, so ~ S 1 . Let ~x,~ y S 1 , so that we have x 1 + x 2 = 0, x 3 =- x 2 , y 1 + y 2 = 0, and y 3 =- y 2 . Then ~x + ~ y = x 1 + y 1 x 2 + y 2 x 3 + y 3 , satisfies ( x 1 + y 1 )+( x 2 + y 2 ) = ( x 1 + x 2 )+( y 1 + y 2 ) = 0+0 = 0, and x 3 + y 3 = (- x 2 )+(- y 2 ) =- ( x 2 + y 2 ). Hence, S 1 is closed under addition. Let ~x S 1 and t R , so that we have x 1 + x 2 = 0, x 3 =- x 2 . Then t~x = tx 1 tx 2 tx 3 , satisfies ( tx 1 )+( tx 2 ) = t ( x 1 + x 2 ) = t (0) = 0, and ( tx 3 ) = t ( x 3 ) = t (- x 2 ) =- ( tx 2 ). Hence, S 1 is also closed under scalar multiplication and hence is a subspace of R 3 . Let ~x S 1 . Then x 1 =- x 2 and x 3 =- x 2 , so any vector ~x S 1 has the form ~x = x 1 x 2 x 3 = - x 2 x 2- x 2 = x 2 - 1 1- 1 . Hence, S 1 = span - 1 1- 1 and since this is clearly linearly independent, we have that - 1 1- 1 is a basis for S 1 . b) S 2 = { ax 2 + bx + c P 2 | abc = 0 } of P 2 . Solution: Observe that ~ y = x 2 + 1 S 2 and ~ z = x S 2 , but ~ y + ~ z = x 2 + x + 1 6 S 2 . Hence, S 2 is not closed under addition and thus is not a subspace of P 2 . 1 2 c) S 3 = { p ( x ) P 2 | p (1) = 0 } of P 2 . Solution: In P 2 the zero vector is the polynomial that satisfies z ( x ) = 0 for all x . Hence z (1) = 0, so ~ S 3 . Let p ( x ) ,q ( x ) S 3 . Then p (1) = 0 and q (1) = 0, so ( p + q )(1) = p (1) + q (1) = 0 + 0 = 0 ....
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assign6_soln - Math 136 Assignment 6 Solutions 1....

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