Find the matrix for the linear transformation that rotates every vector in ℝ squared through an angle of 𝜋 over three.
This kind of rotation takes each vector and moves them to a new position by rotating them about a certain angle. We can use a rotation matrix here. We can rotate the vectors ℝ by 𝜃 using this matrix, cos 𝜃, sin 𝜃, negative sin 𝜃, cos 𝜃.
If we want to rotate about 𝜋 over three, we’re making a 60-degree rotation. We would take cos of 𝜋 over three, sin of 𝜋 over three, negative sin of 𝜋 over three, and then cos of 𝜋 over three again. We know cos of 60 degrees equals one-half. And sin of 60 degrees equals the square root of three over two. Using that information, we plug in what we know.
And we’ll have one-half, square root of three over two, negative square root of three over two, and one-half. This is the matrix we would need to rotate every vector by an angle of 𝜋 over three.