Solve Overdetermined Linear System

Hi,

I have read that an overdetermined system is one with more rows (equations) that columns(unknowns).

I believe that an overdetermined system is typically one that is not solvable, unless one of the equations in the linear system is a linear combination of the other.

However, looking at the code below, it doesn't seem that any of the equations is a linear combination of the others. However, matlab still solves the system once

the script is executed. Any explanation is appreciated.

clear all
% Specify a system where the number of unknowns is less than the number of
% equations
comment='starts here'
disp(comment)
A=[ 2 2 -2; 6 4 4; 10 8 6; 11 15 9];
v=[10 2 8 20]';
C=A\v

[Rena Berman]

(Answers Dev) Restored edit

Answer 1

[Bruno Luong]

For overdetermined system the "\" returns least-square solution, meaning it doesn't solve exactly your system, but returnes the solution that minimizes

norm( A*C - v, 2).

To persuade this is the case, you can multiply A*C and verifies it does not match v

>> A*C

ans =

    9.9655
    1.8368
    8.1255
   19.9812

>> norm(A*C-v, 2)

ans =

    0.2096

This is the best "MATLAB" can do. You can try any other C, and you won't ever get any better than 0.2096.