The result of reflecting across the line y=-x and then rotating 330 degrees counterclockwise around the origin is an isometry T. Represent T by a 2x2 matrix.

Could someone explain how to do this?

To represent the isometry T as a 2x2 matrix, we need to find the matrix that represents the composition of the reflection across the line y=-x followed by the rotation of 330 degrees counterclockwise around the origin.

Let's break down the process step by step:

Step 1: Reflection across the line y=-x
To perform a reflection across the line y=-x, the x-coordinate and y-coordinate of every point will be swapped. In other words, if we have a point (x, y), after reflection it will become (-y, -x).

Step 2: Rotation of 330 degrees counterclockwise
To rotate a point counterclockwise around the origin, we can use the following formulas:

x' = x * cos(θ) - y * sin(θ)
y' = x * sin(θ) + y * cos(θ)

Where (x', y') are the coordinates of the point after rotation, (x, y) are the coordinates before rotation, and θ is the angle of rotation.

In this case, the angle of rotation is 330 degrees. However, we need to convert it to radians since trigonometric functions in most programming languages work with radians. The conversion is as follows:

θ (radians) = 330 * π / 180

Step 3: Composition of reflection and rotation
To find the resulting matrix, we need to apply the reflection first and then the rotation. Let's assume that the reflection matrix is R and the rotation matrix is O.

To apply the reflection matrix, we multiply it by the coordinate vector (x, y) using matrix multiplication:

[x' y'] = R * [x y]

Similarly, to apply the rotation matrix, we multiply it by the result of the reflection:

[x'' y''] = O * [x' y']

The resulting matrix T is the one that performs the same transformation as the composition of the reflection and rotation. It maps the point (x, y) to (x'', y'').

By substituting the formulas for reflection and rotation above, we can find the values of the matrix elements.

However, since you specifically asked for the matrix representation, I will provide you with the final answer.

The matrix representation of the isometry T that results from reflecting across the line y= -x and then rotating 330 degrees counterclockwise around the origin is:

T = [ [ -√3/2 -1/2] [ 1/2 -√3/2 ] ]