Introduction:
In this problem, we are given a linear transformation T on R^2 which represents a reflection about the line y = π/e x along the line y = -e/π x. We need to find the matrix representation of T with respect to the basis {(sin √2π), (0, 7)} and calculate its trace.

Matrix Representation of T:
To find the matrix representation of T, we need to determine the images of the basis vectors under T. Let's start by finding the image of the first basis vector (sin √2π).

Image of the First Basis Vector:
The line y = π/e x intersects the x-axis at the point (e/π, 0). The reflection of the point (sin √2π, 0) about the line y = π/e x will be the point on the line y = -e/π x that is equidistant from the line y = π/e x. Let's denote this point as P.

The distance between P and the line y = π/e x is the perpendicular distance from P to the line y = π/e x. Using the formula for the distance between a point and a line, we can calculate this distance as:

d = |(-e/π)(sin √2π) - (π/e)(0) + π/e * e/π| / √((-e/π)^2 + (π/e)^2)

Simplifying the expression, we get:

d = |-(e/π)(sin √2π) + 1|

Since the point P is equidistant from the lines y = π/e x and y = -e/π x, the distance between P and the line y = -e/π x is also d.

Therefore, the image of the first basis vector under T is (-e/π)(sin √2π) + 1.

Image of the Second Basis Vector:
The line y = π/e x intersects the y-axis at the point (0, π/e). The reflection of the point (0, 7) about the line y = π/e x will be the point on the line y = -e/π x that is equidistant from the line y = π/e x. Let's denote this point as Q.

The distance between Q and the line y = π/e x is the perpendicular distance from Q to the line y = π/e x. Using the formula for the distance between a point and a line, we can calculate this distance as:

d = |(-e/π)(0) - (π/e)(7) + π/e * 0| / √((-e/π)^2 + (π/e)^2)

Simplifying the expression, we get:

d = |-(π/e)(7)| / √((-e/π)^2 + (π/e)^2)

Since the point Q is equidistant from the lines y = π/e x and y = -e/π x, the distance between Q and the line y = -e/π x is also d.

Therefore, the image of the second basis vector under T is -(π/e)(7).

Matrix Representation of T:
Now that we have determined the images of the basis vectors under T, we can write