We saw that the solution to the interpolation problem is unique provided all interpolation points are distinct. As we can see geometrically with linear interpolation, the conditioning gets worse as the interpolation points move closer. Try to hold up a ruler with two fingers positioned at the end, moving your fingers closer to each other.
The conditioning of the least squares problem can be seen geometrically. The solution space is spanned by the columns of the overdetermined linear system we have to solve. If the columns are perpendicular to each other, the solution space is very well defined and the problem is well conditioned. If the vectors are close to being linearly independent, small changes in the spanning vectors will lead to huge changes in the positioning of the solution space and the problem is ill conditioned.
Below is the illustration of the example where equidistant spacing of the interpolation points produces a bad approximation.
This example motivates the next chapter.