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I'm reading Axler's Linear Algebra Done Right. There is a nice proof on page 47 that a system of homogeneous linear equations with fewer equations than unknowns must have a nontrivial solution.

However, the more common formulation taught in grade school is that inhomogeneous systems with fewer equations than unknowns have an infinite number of solutions. I can't figure out how to prove this from the theorems Axler has introduced up to this point and, oddly, also can't find a proof online. Is this even true? Does anyone have pointers for how to prove it?

This question has a proof for the case where the corresponding homogeneous solution has a nontrivial solution. It would be nice if I could prove it without this assumption.

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  • $\begingroup$ Multiply all values of the non-trivial solution of the homogeneous by any number you want and add that to a solution of the inhomogeneous. $\endgroup$
    – Hellen
    Commented Sep 23, 2017 at 6:07
  • $\begingroup$ @Hellen I linked that answer in my question; I would like a proof that doesn't use the nontrivial solution of the homogeneous (or assume there is one). $\endgroup$
    – user219923
    Commented Sep 23, 2017 at 6:12

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There is no theorem in Linear Algebra Done Right that an inhomogeneous system of equations with fewer equations than unknowns has an infinite number of solutions because that result is not true. Consider, for example the following system of linear equations: $$ x + y + z = 1\\ x + y + z = 2 $$ Obviously this system of equations has no solutions. This is a system of two equations in three unknowns. Thus even though there are fewer equations than unknowns, there are no solutions.

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You also need the equations to be consistent (i.e., in order not to get incompatible equations like $x=0$, $x=1$). You should take a look at the Rouché-Capelli theorem. The number of solutions (none, one, infinite) is determined by the comparison of the ranks of the matrix of the homogeneous system and of the (augmented) matrix of the complete system.

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Gaussian elimination can yield at most as many pivots as there are equations. So if there are more variables than there are equations, it follows that in the solution to the homogeneous system, at least one variable must be free. Hence there are infinitely many solutions.

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  • $\begingroup$ Thanks; however Axler hasn't introduced Gaussian elimination (or matrices at all) yet so I'd prefer a proof that doesn't use it, if possible. $\endgroup$
    – user219923
    Commented Sep 23, 2017 at 6:13
  • $\begingroup$ I mean, the same argument works with eqns - you just have to describe the steps of Gaussian elimination. Suppose there exists a homogeneous linear system $\{f_m(x_1,...,x_N) = \sum_{n=1}^N a_{mn}x_n =0\}_{m=1}^M.$ with $M<N.$ If $a_{m1}=0$ for all $m$, then c$e_1$ is an infinite family of solutions. If not, let $m^*$ be the first $m$ with $a_{m1}\neq 0$. Then for all $m\neq m^*$, define $\{g_m(x_2,...,x_N) = f_m(-\frac{1}{a_{m^*1}}\sum_{n=2}^N a_{m^*n}x_n,x_2,...x_N) =0\}$. Then you have a new system. Eventually there will be an $x_n$ with all zero coefficients, or you'll hit the last eqn $\endgroup$
    – user474330
    Commented Sep 23, 2017 at 6:44

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