Expressing cartesian unit vectors in terms of plane polar unit vectors to prove that former doesn't depend on position

How do I now express cartesian unit vectors in terms of polar unit vectors to show that they are independent of the $r$ and $θ$?

If you express the cartesian unit vectors in terms of polar unit vectors that means that they are not independent of each other. The unit vectors $i$ and $j$ are independent and so is $hat{r}$ and $hat{theta}$. But one set is not independent of other.

To find $i$ in terms of $r$ and $theta$ by eliminating $j$ from two equation. As follows

$$cos(theta)hat{r}=cos^2(theta)hat{i}+costhetasin(theta)hat{j}$$ $$sin(theta)hat{theta}=-sin^2(theta)hat{i}+cos(theta)sin(theta)hat{j}$$ substracting:

$$hat{i}=cos(theta)hat{r}-sin(theta)hat{theta}$$ similear procedure for $hat{j}$.

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If you want to show the linear independence of $hat{r}$ and $hat{theta}$ then you can show they are perpendicular to each other: $$hat{r}cdothat{theta}=0$$ and for unit vectors $hat{r}cdothat{r}=1=hat{theta}cdot hat{theta}$.