A linear regression is a method to fit a linear function to a set of points (observations) minimizing the least-squares error.
Now imagine your heatmap indicating a shape where you would assume a vertical line fitting best. Just turn your heatmap 10 degrees counter clock-wise and you have it.
Now how would a linear function supposed to be defined which is vertical? Exactly, it is not possible.
The result of this little thought experiment is that you confuse the purpose of linear regression and what you most likely want is - as indicated already by Gavin Simpson - the 1st principal component vector.