12.6: Over-Determined Systems of Equations

Last updated
Save as PDF

Page ID: 85172

Carey Smith
Oxnard College

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Consistent Equations

Simple examples of consistent over-determined system of equations:

One equation is a multiple of another equation
One equation is the of 2 other equations

The rank of a consistent set of equations is equal to the number of unknowns.

Example \(\PageIndex{1}\) "Overdetermined" consistent system: 3 equations, 2 unknowns

x = -1 : 0.2 : 4;
y1 = 5 - 1*x;
y2 = -1 + 2*x;
y3 = 2 + 0.5*x;

The 3 lines intersect in 1 point, so these equations are consistent.

Graphical intersection at (2, 3)

%% Matrix reformulation:

% Put the variables on the left and the constants on the right.
% 1*x + y = 5
% -2*x + y = -1
% -0.5*x + y = 2

% A*xy = B
A = [ 1 1
-2 1
-0.5 1]
B = [ 5
-1
2]
rank(A) % 2
xy = A\B % Matrix algebra solution = (2, 3)

Solution

Add example text here.

.

Exercise \(\PageIndex{1}\) Over-Determined Consistent System of Equations 1

Over-Determined consistent system of equations of 4 equations in 3 unknowns.

A3 = [3 2 5
4 5 -2
1 1 1
2 -4 -7]
% X3 = [x y z]'
B3 = [22
8
6
-27]
Use Left matrix-division to solve this system of equations for X3
Check your solution by computing this:
X3ck = A3*X3

The result should be close to B3.

Answer: Add texts here. Do not delete this text first.

Inconsistent Equations

Example \(\PageIndex{2}\) "Overdetermined" inconsistent system: 3 equations, 2 unknowns

x = -1 : 0.2 : 4;
y1 = 5 - 1*x; y2 = -1 + 2*x; y3 = 1.6 + 0.5*x;

% The 3rd equation has been modified, so the 3 lines do not intersect in a single point.

Measurement Errors were added to the right-hand-side values of the consistent version of this exercise. Extra measure ments are often taken to help average-out measurement errors.

These these equations are inconsistent. Rather, there are 3 intersection points of each pair of lines.

%% We can still use left-division (Gaussian elimination)
% to find a good approximate solution.
% Matrix formulation
% 1*x + y = 5
% -2*x + y = -1
% -0.5*x + y = 1.6]
% xy = [x
% y]
% A*xy = B]
A = [ 1 1
-2 1
-0.5 1]
B = [ 5
-1
1.6]
rank(A) % 2
xy = A\B % Solution = (2, 2.87)

Solution

Add example text here.

Exercise \(\PageIndex{2}\) Over-Determined Inconsistent System of Equations 2

Over-Determined inconsistent system of equations of 4 equations in 3 unknowns.

Measurement Errors were added to the right-hand-side values of the consistent version of this exercise. Extra measure ments are often taken to help average-out measurement errors.

A4 = [3 2 5
4 5 -2
1 1 1
2 -4 -7]
% X4 = [x y z]'
B4 = [22+1
8-0.5
6+1
-27-1]

Use Left matrix-division to solve this system of equations for X4
Check your solution by computing this:
X4ck = A4*X4

The result should be somewhat close to B4, but will not match because these equations are inconsistent.

Answer: Add texts here. Do not delete this text first.