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. [br][br]In this graphic there is a lot of Theory happening: [br][br]1) You use the lower case y hat, beta zero hat, and beta 1 hat[br] -to estimate the Y =xB + e[br][br]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, [br]Square Matrix's, Bases, Subspaces and Dimension, Linear Independence, degrees of freedom, ... , etc.[br][br]3) Hopefully this is correct as possible - there is room for mistake or miss interpretation.[br][br]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. [br][br]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 )