Linear Systems: definition of linear systems or simultaneous equations. We look at systems with the same number of equations as unknowns or more equations than unknowns.

Linear Systems with a Unique Solution

Unique Solutions: solving linear systems of equations with the same number of linearly independent equations as variables.

Example 1: using MATLAB tools to solve a system of equations with three different methods. We focus on using the \ operator for the greatest speed.

Example 2: we rewrite a system of equations into a standard form to facilitate the writing of the “A” matrix and “b” column vector. We then solve the system.

Linear Independence: proving a set of linear equations has a unique solution by showing it has the same number of LI equations as variables. More formal definition of LI.

Example 3: we test a system of equations for LI and show that there is not a unique solution.

Importance of Linear Systems: how do we use linear systems with unique solutions in engineering problems

Overdetermined Systems

Overdetermined Systems: determines the “best fit” line, plane, etc. to go through observed or empirical data points. This is similar to curve fitting from before but can do a better job with multiple independent variables.

Example 4: computing the regression on a small collection of observed points.

Example 5: computing the linear regression on a large sample of points and finding the plane that best passes through the data to make estimates as to the value of the dependent variable.

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