Systems of equations and linear dependence
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Systems of equations and linear dependence
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Every system of simultaneous equations can be interpreted geometrically. For example, corresponding to the equations: [Equation goes here - download the original to see it.] We can view the solution, x = 2, [Equation goes here - download the original to see it.] , as the point of intersection of the two lines [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The system of simultaneous equations will have a unique solution if the graphical representation shows the two lines crossing over to give a point of intersection. However, the graph makes it clear that this need not always be the case. When the two lines corresponding to two equations are parallel there is no solution to the set of simultaneous linear equations:- [Diagram goes here - download the original to see it.] Parallel lines do not give a point of intersection. In two dimensions two lines are parallel when one is a multiple of another. [Equation goes here - download the original to see it.] When one equation is a combination of multiples of other equations that equation is said to be linearly dependent on the others. The entire set is linearly dependent. Thus, only linearly independent systems of simultaneous equations have unique solutions. This also applies in three and in n dimensions. We should also explore the visual interpretations of what happens in three dimensions when a system of three equations has and has not a unique solution. Each equation in the system defines a plane in 3-dimensional space. Thus, for there to be a unique solution, all three planes must intersect at a unique point [Diagram goes here - download the original to see it.] The diagram makes it clear that having a unique point of intersection is the exception rather than the rule. One plane may be parallel to another. Alternatively, each plane may intersect each other to give a line, but the lines may not intersect uniquely [Diagram goes here - download the original to see it.]
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Contents of
Systems of equations and linear dependence
1 Systems of equations and linear dependence
2 Testing for linear independence
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