Linear inhomogeneous system of equations/Elimination/2x+5y+2z-v, 3x-4y+u+2v, 4x -2z+2u/Example

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We have a look at the homogeneous version of example, so we consider the homogeneous linear system

over . Due to

the solution set is a linear subspace of . We have described it explicitly in example as

which also shows that the solution set is a vector space. With this description, it is clear that is in bijection with , and this bijection respects the addition and also the scalar multiplication (the solution set of the inhomogeneous system is also in bijection with , but there is no reasonable addition nor scalar multiplication on ). However, this bijection depends heavily on the chosen "basic solutions“ and , which depends on the order of elimination. There are several equally good basic solutions for .