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

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

Happy Holiday, friends.

After undestanding the definition of mean value, variance and covariance,
I would like to share how to combine linear transformation with what we've learned.

Linear transformation tells how statistics (such as mean value) varies after making a specified action to a dataset. So, if all values of a dataset are changed in a regular (linear) manner (For example, what is the mean value of income if a government agrees to add 1,000 dollars for every citizen in a country?), then we can apply the concept of linear transformation to efficiently get the adjusted statistics.

Here, there are two linear transformations: shift and scale.

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

Shift:
Because of the national holiday, the government decides to add 1000 dollars to every citizen.

income_shift_data = [24000, 51000, 41000]
E[income_shift_data] = (24000+51000+41000)/3
= 38666.67 = 37666.67 + 1000 = E[income_data] + 1000

By observing the change of the value, we got a relationship between the orginal dataset and the shifted dataset, that is, E[income_shift_data] = E[income_data] + 1000.

Having the formular be more general, it as follows:
E[D+a] = E[D] + a, where a is the number of shifted value.

And let's see how linear transformation of scale works tomorrow!
Have a good day.