transpose(A)(b-Ax) = 0 (The zero vector because b-Ax is a vector)
The above equals:
transpose(A)(b) – transpose(A)Ax =0
transpose(A)(b) = transpose(A)Ax
If the matrix is full column rank or of course full rank we can multiply the left inverse of transpose(A)A and get the identity matrix.
(transpose(A)A)¯¹*transpose(A)Ax = (transpose(A)A)¯¹*transpose(A)(b)
Which leaves us with the identity matrix on the left side.
x= (transpose(A)A)¯¹*transpose(A)(b)
