Matrix Multiple Regression Analysis - Actually it's pretty complicated ; I believe the " y " hat equation is used for calculating the Residuals or Variance " e " , while the" y "vector is used to calculate Beta for the Main Y = XB + e equation.
In this graphic there is a lot of Theory happening:
1) You use the lower case y hat, beta zero hat, and beta 1 hat
-to estimate the Y =xB + e
2) Theory of Linear Algebra, Theory of Multiple Regression Analysis, Applied Linear Algebra, Matrix Multiplication, Dot Product, Statistics, Sums and Averages, Vector addition and subtraction, Pythagorean Theorem, Inverse Matrix's,
Square Matrix's, Bases, Subspaces and Dimension, Linear Independence, degrees of freedom, ... , etc.
3) Hopefully this is correct as possible - there is room for mistake or miss interpretation.
4) In terms of myself putting i=j on the X - Matrix - What I am referring too is that the Vectors are Column Vectors which are in the j direction, In addition too the adaption of indices transformations.
5) The ei's are Independent of the Y i's and i does not equal j for the error term or the Variance term ( e )