Question

Suppose T is a transformation from ℝ2 to ℝ2. Find the matrix A that induces T if T is:

a) reflection over the line y=−1/3x
b) rotation by 1/3π

a) A =

b) A =

Answer 1

To find the matrix A that induces transformation T, we consider how T acts on the standard basis vectors in ℜ2. For a reflection over y = -1/3x, A = [-1/3, 0; 0, -1]. For a rotation by 1/3π, A = [cos(1/3π), -sin(1/3π); sin(1/3π), cos(1/3π)].

Step-by-step explanation:

To find the matrix that induces the transformation T, we need to consider how T acts on the standard basis vectors (1,0) and (0,1) in ℜ2.

a) For a reflection over the line y = -1/3x, T(-1/3x, y) = (-1/3x, -y). Thus, the matrix A for this transformation is:

| -1/3 0 |

| 0 -1 |

b) For a rotation by 1/3π, we need to use the rotation formulas:

x' = x cos(θ) - y sin(θ)

y' = x sin(θ) + y cos(θ)

With θ = 1/3π, the matrix A for this transformation is:

| cos(1/3π) -sin(1/3π) |

| sin(1/3π) cos(1/3π) |