# 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 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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