1. ## Linear Matrix Transformations

Hi Everyone,
I would be really grateful if you could help me out on these tricky questions.

Let f be the linear transformation represented by the matrix M =
(a) state what effect f has on areas, and whether f changes orientation.
(b) Find the matrix that represents the inverse of f.
(c) (i) Use the matrix that you found in part (b) to find the image f() of the unit circle under f, in the form
ax² + bxy + cy² = d
where a, b, c and d are integers.
(ii) What is the area enclosed by f()?

Anyone who can shed some light will receive my eternal gratitude!

2. Originally Posted by looking0glass
Hi Everyone,
I would be really grateful if you could help me out on these tricky questions.

Let f be the linear transformation represented by the matrix M =
Since you have not specified the matrix $\displaystyle M$ I can only contribute some generalities; but I assume that a specific matrix $\displaystyle M$ needs to be considered in this exercise.
(a) state what effect f has on areas, and whether f changes orientation.
Consider the absolue value and the sign of the determinant of $\displaystyle M$
(b) Find the matrix that represents the inverse of f.
Well, as I wrote, you need to tell us what $\displaystyle M$ happens to be.

(c) (i) Use the matrix that you found in part (b) to find the image f() of the unit circle under f, in the form
ax² + bxy + cy² = d
where a, b, c and d are integers.
If $\displaystyle (x,y)\in f(\ell)$ the inverse image $\displaystyle (x',y'):=f^{-1}(x,y)$ must satisfy $\displaystyle x'^2+y'^2=1$. Now express $\displaystyle x',y'$ in terms of $\displaystyle x,y$ according to your solution of (b).

(ii) What is the area enclosed by f()?
See (a).