More on Matrices Elimination

The well-known Gauss method of elimination is older than the theory of vector spaces. It will be outlined briefly, only to brush up the reader s memory. Let us again forget, for a moment, what we know about vectors and matrices. Let us have a set of M linear equations in unknowns x, —,  

By any of the operations, the set of equations is transformed, but the solutions X, —, Xn, if they exist, remain the same. This holds true, consequently, as well for an arbitrary sequence of elementary operations. Observe also that we can write the variables in an arbitrary order. 

The strategy is as follows. One finds some A 0 the m-th equation is placed as first. One multiplies the equation by 1M , , getting coefficient 1 at x , one then subtracts Am. -times the equation firom each of the m -th equations 

For a better visualisation, let us arrange the variables in the order, and the remaining ones. Then the equations read 

This is (a slightly modified version of) the famous Gauss elimination. The solvability analysis that follows is well-known for example if some 0 for m K then the equations are not solvable. If the original a, , equal zero (a homogeneous system of equations), they remain such by the operations, hence also a = = a = 0 and the equations are solvable, at least by jc, = — = jc = 0. Clearly, K N. On the other hand if N M then after follow necessarily certain, , zs K M if a, = = = 0 then setting for example 

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