Let f be the invertible linear transformation represented by the matrix
A = (-2 -3
2 -3)
Find in terms of x and y the equation of the image f(C) and of the unit circle C
To find the equation of the image f(C), we need to apply the linear transformation represented by the matrix A to the points on the curve C.
The unit circle C is defined by the equation x^2 + y^2 = 1.
We can write this equation in matrix form as [x, y] • [x, y]ᵀ = 1.
To find the equation of the image f(C), we can simply apply the matrix A to each point [x, y] on the unit circle.
Let's find the image of the point [x, y] on the unit circle:
[f(x, y)] = A • [x, y]
= [(-2 -3) • (x)
(2 -3)] (y)
= [-2x - 3y
2x - 3y]
Therefore, the equation of the image f(C) can be written as:
(-2x - 3y)^2 + (2x - 3y)^2 = 1
Simplifying this equation, we get:
4x^2 + 9y^2 + 12xy - 4x - 6y = 1
So, the equation of the image f(C) in terms of x and y is:
4x^2 + 9y^2 + 12xy - 4x - 6y - 1 = 0.
Note: This equation represents an ellipse, which is the image of the unit circle under the transform f.