Question

The linear transformation T(x⃗) = [0.6 0.8; 0.8 -0.6] represents a reflection about a line L. Find the equation of line L (in the form y = mx).

a) y = -0.6x
b) y = 0.6x
c) y = -0.8x
d) y = 0.8x

Answer 1

The equation of the line representing the reflection described by the given linear transformation is y = 1.6667x.

Step-by-step explanation:

The linear transformation T(x⃗) = [0.6 0.8; 0.8 -0.6] represents a reflection about a line L. To find the equation of line L, we need to find the slope m. In a reflection, the line of reflection is perpendicular to the line connecting a point and its reflected image. Since the transformation matrix is a reflection, the slope of line L will be the negative reciprocal of the slope between two points on the line L. Let's take two points (x1, y1) and (x2, y2) online L. The slope between these two points is (y2 - y1) / (x2 - x1). The negative reciprocal of this slope will give us the slope of line L.

1. Choose two points online L, let's say (x1, y1) = (1, 0.6) and (x2, y2) = (0, 0).
2. Calculate the slope between these two points: m = (0 - 0.6) / (0 - 1) = -0.6.
3. The negative reciprocal of m will be the slope of line L: -1 / (-0.6) = 1.6667.

Therefore, the equation of line L is y = 1.6667x.