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.

Question

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.

Allison Pena · Accepted Answer

A transformation of the form   A      [                            x                                      y                      ]   is linear: it maps the origin to the origin; that&#039;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&#039;s the linear system you are looking for:      [                            4          −          1                          2                                      2                          7          −          1                      ]        [                            x                                      y                      ]    =  −      [                            14                                      42                      ]    .

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

Answered question

Answer & Explanation

New Questions in Linear algebra