Linearly independent coordinates Consider the coordinate transformation $$(x,y)\to (a(x,y),b(x,y)).$$

I'm told that in order for $a(x,y)$ and $b(x,y)$ to be linearly independent, we must have $$\begin{vmatrix} \frac{\partial a}{\partial x} &amp; \frac{\partial a}{\partial y} \\ \frac{\partial b}{\partial x} &amp; \frac{\partial b}{\partial y}\end{vmatrix} \neq 0.$$

I'm not sure how to justify why this condition ensures linear independence. 

For a set of vectors $\{v_1,\ldots,v_n\}$, I know that these are linearly independent if \begin{align*} \lambda_1v_1+\cdots+\lambda_nv_n=0 \implies \lambda_1=\cdots =\lambda_n=0.\end{align*}

I'm unsure of how to apply this idea to get the condition above though.
The condition is about Jacobian determinant and assures that the transformation is locally invertible, thus the gradient vectors of $a$ and $b$ must be linearly independent.
The condition is about Jacobian determinant and assures that the transformation is locally invertible, thus the gradient vectors of $a$ and $b$ must be linearly independent.