The least square method was first put forward by Gauss, who used this method to solve astronomical problems, especially to determine the celestial trajectories of some planets and comets. The elliptical trajectory of such celestial bodies is determined by five parameters. In principle, measuring its position five times is enough to determine its entire trajectory. However, due to the measurement error, the trajectory determined by five measurements is extremely unreliable. On the contrary, it is necessary to measure many times, eliminate the measurement error by least square method, and get more accurate ballistic parameter values. The least square method approximately reduces the high-dimensional space problems caused by dozens or even hundreds of observations to the five-dimensional parameter space of the elliptical trajectory model.

If you want to know the monthly rainfall in a place, one month's observation is definitely not enough. Any month may be unusually sunny or rainy. Instead, people should study it for months or at least a year or even a decade, and average all the data. The average result may not be completely consistent with any specific month, but intuitively, the standard rainfall figure given by this result will be much more accurate than that obtained by studying for only one month. This principle is universal in the fields of observation and experimental science. It eliminates measurement errors and random fluctuations through multiple measurements. The carpenter's motto "Look before you leap" is also an example of this common sense.

In the case of rainfall, we use a number to represent or approximate the effect of the whole measurement data to some extent. More generally, low-dimensional objects are often used to approximate high-dimensional objects for various theoretical and practical reasons. This method can be used for the following tasks, such as eliminating errors or ignoring irrelevant details, extracting signals from interference data or looking for trends, reducing a large number of data to a manageable number or replacing complex functions with simple approximations. We don't expect this approximation to be very accurate. In fact, in many cases, it does not need to be very accurate. But despite this, we still hope that it can maintain the similarity with the original data. In the field of linear algebra, we hope to project a vector from a high-dimensional space to a low-dimensional subspace. One of the most common and convenient ways to accomplish this task is the least square method.