🚀 go-pugleaf

CORRECTION to my previous post:

Fact #1

An nxn linear system Ax=b has a unique solution if
and only if A is nonsingular.

Fact #2

An mxn (where m .ne. n) linear system Ax=b has either 
no solution (termed "inconsistent") or an infinite number 
of solutions.

Fact #3

The best you can do is a "least squares" solution.
Notation: A' = transpose of A
          A^ = inverse of A 
          ||x|| = Euclidean norm of the vector x
Say you want to solve Ax=b where A is nx3
==> A'Ax = A'b  (these are the "normal equations")
    If the columns of A are linearly independent, then the
    3x3 matrix A'A is nonsingular.
==> x = (A'A)^A'b, i.e., x is the unique solution to the 
    normal equations and the unique least squares solution
    to Ax=b.
    
Definition of a least squares solution:
 x is least squares solution if and only if
 ||Ax - b|| .le. ||Az - b|| for all other vectors z.
                   ^ ---> NOTE this correction.