Q: (x, y) → (2x-3y, -4x - 6y).
(a) Write down the matrices of P and Q. (b) What is the image of (-2,-3) under the transformation Q?
(c) Obtain a single transformation representing the transformation Q followed by P.
(d) Find the image of (1,4) when transformed by Q followed by P.
(e) Find the image P\(^1\) of the point (-√2,2√2) under an anticlockwise rotation of 225° about the origin.
(a) The matrix representation of a linear transformation can be obtained by writing the image of the standard basis vectors (1,0) and (0,1). The matrix of P is given by:
P = [[-3, 6], [4, 1]]
Similarly, the matrix of Q is given by:
Q = [[2, -3], [-4, -6]]
(b) To find the image of a point (-2,-3) under the transformation Q, we first write the point as a column vector:
(-2,-3) = [-2, -3]^T
Next, we multiply the matrix Q with the vector:
Q * [-2, -3]^T = [2, -3] * [-2, -3]^T = [-12, 18]^T
So the image of (-2,-3) under the transformation Q is (-12, 18).
(c) To find the single transformation representing the transformation Q followed by P, we multiply the matrices Q and P:
Q * P = [[2, -3], [-4, -6]] * [[-3, 6], [4, 1]] = [[36, -18], [-24, -30]]
So the single transformation representing the transformation Q followed by P is given by the matrix [[36, -18], [-24, -30]].
(d) To find the image of (1,4) when transformed by Q followed by P, we first write the point as a column vector:
(1, 4) = [1, 4]^T
Next, we multiply the matrix representing Q followed by P with the vector:
[[36, -18], [-24, -30]] * [1, 4]^T = [36, -18] * 1 + [-24, -30] * 4 = [36, -18] + [-96, -120] = [-60, -138]^T
So the image of (1,4) when transformed by Q followed by P is (-60, -138).
(e) To find the image P^1 of the point (-√2,√2) under an anticlockwise rotation of 225° about the origin, we can first rotate the point by 225° in the counterclockwise direction and then apply the transformation P.
The counterclockwise rotation of 225° can be represented by the matrix:
R = [[cos(225), -sin(225)], [sin(225), cos(225)]] = [[-√2/2, √2/2], [-√2/2, -√2/2]]
Next, we multiply the matrix R with the vector representing the point (-√2,√2):
R * [-√2, √2]^T = [[-√2/2, √2/2], [-√2/2, -√2/2]] * [-√2, √2]^T = [-√2/2 * -√2 + √2/2 * √2, -√2/2 * -√2 - √2/2 * √2]^T = [√2, -√2]^T
So the image of the point (-√2,√2) under the counterclockwise rotation of 225° is (√2, -√2). Finally, to find