assign6_soln

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

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Unformatted text preview: Math 136 Assignment 6 Solutions 1. Find a spanning set for the kernel and range for each of the following linear mappings. a) f ( x 1 ,x 2 ,x 3 ) = ( x 1 ,x 2 ,x 1 + x 2- x 3 ) Solution: If ~x R 3 is in ker( f ), then (0 , , 0) = f ( x 1 ,x 2 ,x 3 ) = ( x 1 ,x 2 ,x 1 + x 2- x 3 ) Hence, we must have x 1 = 0, x 2 = 0, and x 1 + x 2- x 3 = 0, which implies that x 3 = 0. Thus, if ~x ker( f ), then ~x = ~ 0. So ker( f ) = span { ~ } . For any ~x R 3 , we have f ( x 1 ,x 2 ,x 3 ) = x 1 x 2 x 1 + x 2 + x 3 = x 1 1 1 + x 2 1 1 + x 3 1 Thus Range( f ) = span 1 1 , 1 1 , 1 . b) f ( x 1 ,x 2 ,x 3 ) = (0 ,x 1- x 2 ) Solution: If ~x R 3 is in ker( f ), then (0 , 0) = f ( x 1 ,x 2 ,x 3 ) = (0 ,x 1- x 2 ) Hence, we must have x 1- x 2 = 0, which implies x 1 = x 2 . Thus, if ~x ker( f ), then ~x = x 1 x 1 x 3 = x 1 1 1 + x 3 1 So ker( f ) = span 1 1 , 1 ....
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assign6_soln - Math 136 Assignment 6 Solutions 1. Find a...

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