Image of geometrical figures by linear applicationWhen the application is a rotation, symetry, projection, it is easy to find without any calculus the image of figures such as squares, circles, and so on. But how do you find these images when the application is not 'geometrically obvious'?For example consider the application f ( x , y ) : R 2 → R 2 : ( x , y ) → ( 2 x + y , x + 3 y ), then what is the image of the square of summits ( ± 1 , ± 1 )? And the image of the circle x 2 + y 2 = 1?

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Image of geometrical figures by linear applicationWhen the application is a rotation, symetry, projection, it is easy to find without any calculus the image of figures such as squares, circles, and so on. But how do you find these images when the application is not &#039;geometrically obvious&#039;?For example consider the application   f  (  x  ,  y  )  :            R      2        →            R      2        :  (  x  ,  y  )  →  (  2  x  +  y  ,  x  +  3  y  ), then what is the image of the square of summits   (  ±  1  ,  ±  1  )? And the image of the circle       x    2    +      y    2    =  1?

Antwan Wiley · Accepted Answer

Step 1For the square: a linear map takes parallel lines to parallel lines, so the image of a square is a parallelogram. You can easily find the coordinates of its vertices.For the unit circle: the circle can be written as             x        T        x    =  1     ,, where       x    =      (                            x                                      y                      )       ..Step 2Its image will be   (  A      x        )    T    (  A      x    )  =  1     ,, where   A  =      (                            2                          1                                      1                          3                      )       .If my calculations are correct (please check) this simplifies to             x        T        (                            5                          5                                      5                          10                      )        x    =  1     ,, and the matrix has positive eigenvalues so this is an ellipse.

Consider the application f(x,y):R^2 rightarrow R^2:(x,y) rightarrow (2x+y ,x+3y), then what is the image of the square of summits (pm 1, pm 1)? And the image of the circle x^2+y^2=1?

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