Find the matrix for the linear transformation that rotates every vector in ℝ two through an angle of 𝜋 over four.
This kind of rotation takes each vector and moves them to a new position by rotating them about a certain angle. We can use the rotation matrix here cause what we can do is rotate the vectors ℝ by 𝜃 using this matrix that we have here. And that’s the matrix cos 𝜃, negative sin 𝜃, sin 𝜃, cos 𝜃. If we want to rotate about 𝜋 over four, we’re making a 45-degree rotation. So what we’d do is we would take cos 𝜋 over four, negative sin 𝜋 over four, sin 𝜋 over four, and cos 𝜋 over four.
Well, since we know that cos of 45, because 𝜋 over four is 45, is equal to root two over two and the sin of 45 is equal to root two over two, what we can do is plug these in. And when we do plug these values in, what we’re gonna have is the matrix for the linear transformation that rotates every vector in ℝ squared through an angle of 𝜋 over four. And that matrix is going to be the matrix root two over two, negative root two over two, root two over two, root two over two.