The brief structure leading to the milestone of PCA is as below:

1. Statistical Introduction
2. Transformation of Vectors in Spaces
3. Orthogonal Projectio

We talked linear transformation of shift yesterday, let's look into the scale today.

Assume the original dataset and its mean value is as follows:
income_data = [23000, 50000, 40000]
E[income_data] = (23000+50000+40000)/3 = 37666.67

Scale: Because of the economic miracle, the government decides to have every citizen's income double.
*income_scale_data = [46000, 1000000, 80000]
E[income_scale_data] = (46000+100000+80000)/3
= 75333.33 = 37666.67 * 2 = E[income_data]2

By observing the change of the value, we got a relationship between the orginal dataset and the scaled dataset, that is, E[income_scale_data] = E[income_data]*2.

Having the formular be more general, it as follows:
E[D*b] = E[D]*b, where b is the number of scaled value