The transformation <mtext mathvariant="bold">T</mtext> maps points ( x , y ) of the

orlovskihmw 2022-07-07 Answered
The transformation T maps points ( x , y ) of the plane into image points ( x , y ) such that
x = 4 x + 2 y + 14 y = 2 x + 7 y + 42
Find the coordinates of the invariant point of T. Hence express T in the form
( x y + k ) = A ( x y + k )
where k is a positive integer and A is a 2 × 2 matrix.
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Answers (1)

Allison Pena
Answered 2022-07-08 Author has 14 answers
A transformation of the form A [ x y ] is linear: it maps the origin to the origin; that's not what your T does. For an affine transformation, you need to add the constant term. So your T is of the form
T [ x y ] = [ 4 2 2 7 ] [ x y ] + [ 14 42 ] .
So you have
T v = A v + r .
A fixed point will satisfy
A v + r = v ,
which we may rewrite as
( A I ) v = r .
So that's the linear system you are looking for:
[ 4 1 2 2 7 1 ] [ x y ] = [ 14 42 ] .

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