Rotating vectors with transformations

Question

Let $θ_0$ be a real number. Define $R : \mathbb{R}^2 → \mathbb{R}^2$ by $R(x, y) = ((cos θ_0)x − (sin θ_0)y,(sin θ_0)x + (cos θ_0)y)$.

How do I show that $R$ rotates both $\mathbf{i}$ and $\mathbf{j}$ by the angle $θ_0$ counterclockwise? I know that $R$ should be a linear transformation

Answer 1

The image of the vector $\mathbf i=(1,0)$ is $(\cos \theta_0, \sin \theta_0)$. The inner product of this last vector with vector $\mathbf i$ equals $\cos \theta_0$. And the cross product $\sin\theta_0$. This gives the conclusion for $\mathbf i$.

You can do a similar analysis for $\mathbf j$.