Matrix transformations- please help?
let f be the linear transformation represented by the matrix
M = (4 2)
.......(0 -2)
a) state what effect f has on areas and whether it changes orientation
b) Find the matrix that represents the inverse of f
c) Use the matrix you found in part b to find the image f(c) of the unit circle C under f , in the form
ax^2 + bxy + cy^2 = d
where a, b, c, d are integers
d) what is the area enclosed by f(c)?
My answers;
a)det M = (4*-2) - (2*0) = -8 so f scales areas by factor 8 and changes orientation
b) m^-1 = 1/-8 (-2 -2) ... (1/4 1/4)
..............=..............o 4) = (0 -1/2)
c) 1/16x^2 + 2/16xy + 1/16y^2 = 1
which is;
x^2 + 2xy + y^2 = 16
(ive struggled with this one so if its wrong i would appreciate some help with my working out)
d) the area enclosed by f(C) is 16pi
Thanks
Your answers for parts a) and b) are correct. Let's go through part c):
To find the image f(c) of the unit circle C under f, we need to substitute the equation of the unit circle into the inverse matrix.
The equation of the unit circle C is given by:
x^2 + y^2 = 1
Now, substitute x = (1/4)x' - (1/2)y' and y = x' into the equation:
((1/4)x' - (1/2)y')^2 + (x')^2 = 1
Simplifying:
(1/16)x'^2 - (1/8)x'y' + (1/4)y'^2 + x'^2 = 1
Combining like terms:
(17/16)x'^2 - (1/8)x'y' + (1/4)y'^2 = 1
Thus, the image f(c) of the unit circle C under f is given by the equation:
(17/16)x'^2 - (1/8)x'y' + (1/4)y'^2 = 1
In the requested form:
(ax^2 + bxy + cy^2 = d), we can simplify by multiplying through by 16 to get rid of the fractions:
17x'^2 - 2x'y' + 4y'^2 = 16
Comparing this to the form ax^2 + bxy + cy^2 = d, we can identify a = 17, b = -2, c = 4, and d = 16.
For part d):
The area enclosed by f(c) is determined by the coefficient of x^2 and y^2 in the equation. In this case, a=17 and c=4.
The area enclosed by f(c) is given by the formula: π * |ac|
Therefore, the area enclosed by f(c) is π * |17 * 4| = π * 68 = 68π.
Hence, the area enclosed by f(c) is 68π.
Hopefully, this clarifies the solution for part c) and part d) of the question. Let me know if you need any further assistance!