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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 FBA

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 y-value may
be interpolated for any x-value within the x-data
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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