Find the matrix for the linear transformation which rotates every vector in ℝ two through an angle of 𝜋 over 12.
This kind of rotation takes each vector and moves them to a new position by rotating them about a certain angle. And what we can do with this kind of problem is use a rotation matrix. What we can do is rotate the vectors ℝ by 𝜃 using this matrix. We’ve got cos 𝜃, negative sin 𝜃, sin 𝜃, cos 𝜃. Now, if we want to rotate about an angle of 𝜋 over 12, this is also the same as rotating by 15 degrees. And to do this, what we could take is cos of 𝜋 over 12, negative sin of 𝜋 over 12, sin of 𝜋 over 12, and cos of 𝜋 over 12. Well, what we get if we find the cos of 𝜋 over 12 and the sin of 𝜋 over 12 is root six plus root two over four and root six minus root two over four, respectively.
So now what we can do is substitute these back into our matrix. And if we do this, what we’re gonna get is the matrix root six plus root two over four, root two minus root six over four — and we get this element because if we had negative root six minus root two over four, this is the same as root two minus root six over four — then root six minus root two over four, and finally root six plus root two over four.