 # 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 ## 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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