7 | Simultaneous Equations & Linear Systems
In this section, we focus on analyzing linear systems of equations. Primarily, we look at “solvable” linear systems: those with an equal number of variables and linearly independent (LI) equations. We then look at overdetermined systems — more LI equations than variables.
At the end of this section you should be able to:
Place a system of linear equations into canonical (standard) form
Use matrix algebra methods to determine if a system of equations has zero, one or infinite solutions
For systems with a unique solution, use methods in Matlab to solve for the variables
For overdetermined systems, use linear regression to determine the best-fit line through the data cloud
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.