I was watching a lecture on tensors and the professor said that a defining feature of a vector $v$ is that it transforms under a coordinate transformation $x^{\mu} \rightarrow x^{\mu'}$ as
$$v^{\mu'}(x^{\mu'}) \equiv \frac{\partial x^{\mu'} }{\partial x^\mu} v^{\mu}(x^{\mu}(x^{\mu'})) $$
Basically the last last term on the right hand side is the $x^{\mu}$ coordinates expressed in terms of $x^{\mu'}$ coordinates.
I am trying to understand this in $\mathbb R^2$ for the Cartesian and Polar coordinates. If the $x^{\mu'} \equiv (x,y)$ and $x^{\mu} \equiv (r,\theta)$, then I get
$$\bigg(\begin{matrix}x\\y \end{matrix}\bigg) \equiv \bigg[ \begin{matrix} \cos\theta&-r\sin\theta\\ \sin\theta&r\cos\theta \end{matrix}\bigg] \bigg(\begin{matrix}r\\\theta \end{matrix}\bigg)$$
which doesn't work out to be true. Can you please tell me what I'm missing here?