1. Give a counterexample to show that the given transformation is not a linear t

Question

1. Give a counterexample to show that the given transformation is not a linear transformation. (b) T 2ry y+1 1-6 1 2. Let T(x) 3 12 -24 (a) Find T1 3 (b) Find all preimages x where T(x) 3 2 3. Use a rectangular coordinate system to plot u - and their images under the given transformation T defined by T(x) - x. Describe geometrically what T does to each vector x in IR2 . Find the standard matrix of the given linear transformation from R2 to R2 (a) Counterclockwise rotation through 150° about the origin (b) Reflection in the r-axis, followed by clockwise rotation through 45° about the origin

Explanation / Answer

1 (a) Let (x1,y1)T, and (x2,y2)T be 2 arbitrary vectors in the domain of T.Then,T(x1,y1)+T(x2,y2)=(x1-y1, 2x1 y1)T +(x2-y2, 2x2 y2)T = (x1-y1+ x2-y2, 2x1 y1+2x2 y2)T = (x1+x2+ -y1-y2, 2x1y1+2x2y2)T and T((x1,y1)T+(x2,y2)T) = T(x1+x2,y1+y2)T = (x1+x2-y1-y2, 2(x1+x2)(y1+y2))T= (x1+x2-y1-y2,2x1y1+2x1y2+2x2y1+2x2y2)T. Thus, T(x1,y1)+T(x2,y2) T((x1,y1)T+(x2,y2)T). This means that T does not preserve vector addition. Hence T is not a linear transformation.

(b) Let (x,y)T be an arbitrary vector in the domain of T and let be an arbitrary scalar.Then,T((x,y)T)=   T( x,y)T= (x-2, y+1)T and T(x,y)T = (x-2,y+1)T = (x-2, y+)T so that T((x,y)T) T(x,y)T. This means that T does not preserve scalar multiplication. Hence T is not a linear transformation.

2. We have T(X) = AX, where A =

1

-6

14

-3

12

-24

Then T(1,1,1)T= A(1,1,1)T = (9,-15)T.

Please post the remaining questions again.

1

-6

14

-3

12

-24

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