Linear transformations can be very important in graphics. Also, linear transformations come up whenever you need to solve systems of linear equations, which arise quite often. Finally, they can be useful in further areas of mathematics such as topology.
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Linear programming approach does not apply the same way in different applications. In some advanced applications, the equations used for linear programming are quite complex.
Accurate linear measurement.
The applications are in finding optimum solutions to a linear objective function, subject to a number of linear constraints.
If the two equations are linear transformations of one another they have the same solution.
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