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Linear Algebraic Systems
Objectives
• Define underdetermined systems
• Define overdetermined systems
• Least Squares Examples
Review
Review cont
The system of matrices above is of the form:
Ax = b
and can be solved using MATLAB left division thus , x = A\b
results in a 1*2 matrix of values for F _{BC} and F_{BA}
Review Summary
• A system of two Equations and two unknowns may yield a unique solution.
• The exception is when the determinant of A is equal to zero . Then the system
is said to be singular.
• The left division operator will solve the linear system in one step by
combining two matrix operations
• A\B is equivalent to A ^{1}*B
Graphical Representation of
Unique vs. Singular Systems
Underdetermined Systems
• A system of linear equations is may be undetermined if;
1 The determinant of A is equal to zero
2 The matrix A is not square, i.e. the are more
unknowns than there are equations
Overdetermined Systems
• The converse of an underdetermined system
is an overdetermined system where there are
more equations than there are variables
• This situation arises frequently in
engineering. For example: suppose a linear
relationship is expected between x and y and
there are multiple data points.
Data Distribution of Linear
Phenomena
experimental data
experimental data  
y = mx + b 
The line y = mx + b, that best describes this
data is obtained by the method of least squares
Method of Least Squares
The line that results in the minimum value of J is the least squares linear fit to the data. 
Example
The Overdetermined System
• Once the curve fit is obtained, a yvalue may
be interpolated for any xvalue within the xdata
range (sometimes extrapolation is
possible).
• In the following example, fmins is used to
minimize the sum of the squared residuals
with respect to the slope and intercept .
Flowchart for least.m
Run least.m
Solving The Overdetermined
System, Method II
• Solving the overdetermined system is
carried out the same way as the other linear
algebra solutions using the left division
method
Method II (cont’d)
• If the system is not overdetermined, the
method will not work
Overdetermined system cont.
Program least2.m
Run least2.m
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