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lina-20-11-09-s1

# lina-20-11-09-s1 - Linear Algebra Geometry Solutions to...

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Linear Algebra Geometry: Solutions to Sheet 7 1. We have to check that ( i ) T ( x + y ) = T ( x ) + T ( y ) and ( ii ) T ( λ x ) = λT ( x ). (a) T ( x,y ) = ( x - y, 5 x ) is linear. (i) let x = ( x,y ), y = ( x 0 ,y 0 ), then T ( x + x 0 ,y + y 0 ) = ( x + x 0 - y - y 0 , 5 x +5 x 0 ) = ( x - y + x 0 - y 0 , 5 x + 5 x 0 ) = T ( x,y ) + T ( x 0 ,y 0 ). (ii) T ( λx,λy ) = ( λx - λy, 5 λx ) = λ ( x - y, 5 x ) = λT ( x,y ). (b) T ( x,y ) = ( x - y 2 , 5 x ) is not linear, e.g. T (2 x, 2 y ) = (2 x - 4 y 2 , 10 x ) = 2( x - 2 y 2 , 5 x ) 6 = 2 T ( x,y ) (c) T ( x,y ) = ( x - 1 , 5 x ) is not linear, e.g. T (2 x, 2 y ) = (2 x - 1 , 10 x ) = 2( x - 1 / 2 , 5 x ) 6 = 2 T ( x,y ) (d) T ( x ) = ( x · e 1 ) u 1 + ( x · e 2 ) u 2 is linear (i) T ( x + y ) = (( x + y ) · e 1 ) u 1 + (( x + y ) · e 2 ) u 2 = ( x · e 1 ) u 1 + ( y · e 1 ) u 1 + ( x · e 2 ) u 2 + ( y · e 2 ) u 2 = ( x · e 1 ) u 1 + ( x · e 2 ) u 2 + ( y · e 1 ) u 1 + ( y · e 2 ) u 2 = T ( x ) + T ( y ) (ii) T ( λ x ) = ( λ x · e 1 ) u 1 + ( λ

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lina-20-11-09-s1 - Linear Algebra Geometry Solutions to...

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