An affine transformation of R2 is a function T: R2 rightarrow R2 of the form T(x

Question

An affine transformation of R2 is a function T: R2 rightarrow R2 of the form T(x) = Ax + b, where A is an invertible 2x2 matrix and b epsilon R2. Which of the following statements are correct? T-1(x) = A-lx-A-lb. Affine transformations map straight lines to straight lines. There is no affine transformation that can map a straight line to a circle. Affine transformations map parallel straight lines to parallel straight lines. There exists an affine transformation that maps parallel straight lines to intersecting straight lines.

Explanation / Answer

A,B,C,D are all true. An affine transformation preserves collinearity and parallelness.Therefore any affine transformations m aps lines to lines, making Band C true. Parallel lines go to parallel lines, so D is true. For A, consider T(x) = y = Ax+b, and we want to find T^-1 so thatT^-1(y) =x. A^-1(y-b) = A^-1Ax = x, so T^-1(y) = A^-1(y-b) = A^-1y-A^-1b forall y, as desired, so A follows. E is false, since parallelness is preserved,

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