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I would normally use Gaussian Elimination to solve a linear system. If we have more unknowns than equations we end up with an infinite number of solutions. Are there any real life applications of these infinite solutions? I can think of solving puzzles like Sudoku but are there others?

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Is your question "Where do problems of infinite solutions come up?" or "Where do I use Gaussian Elimination for solving problems with infinite solutions"? Your content suggests the former, the title suggests the latter. – Inquest Jun 8 '12 at 15:34
possible duplicate Purpose of Inverse matrix – Peter Sheldrick Jun 8 '12 at 15:36

I don't think Gaussian elimination is something which is just useful by is a process that turns the ad hoc ways of solving linear equations into an easy to apply algorithm on matrices.

Linear equations define linear spaces. The equation $3x + 2y + z = 0$ defines a plane in $3$-dimensions. If we throw in another equation of a similar form then solving means the same as finding the points of intersection. As you can imagine, the points of intersection could fall into many possibilities:

1) Might not be any points of intersection, the objects might not intersect (say parallel planes).

2) Might be just one point of intersection (say a line intersecting a plane).

3) Might be a whole line of intersections (say a line lying in a plane or two planes intersecting).

4) Might be a whole plane of intersections (say two of the exact same plane intersecting).

The existence of an infinite amount of solutions tells you that either situation 3 or situation 4 is happening. So really solving linear equations tells us a bit about geometry too.

Many things involve linear equations and having the existence of infinitely many solutions is not a bad thing (it usually means more choice for whoever needs to use the solutions).

It is not a case of being told "this is what you use linear equations for" and leaving it at that. As usual with maths it is a case of "I will create my own ways to use linear equations to model situations".

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In Chapter 9 of the book Higher-Order Perl there is a structured diagram-drawing program that works by generating a system of linear equations that must be satisfied by the various components of the diagram, and then solving the system to determine the location of each component. The system might be underconstrained, in which case not all the features of the diagram can be located and drawn. If this happens, the program can still go ahead and draw part of the diagram anyway, corresponding to the part that it could solve, and omit only the underconstrained part.

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One important application is this: Given the corner points of a convex hull $\{\mathbf v_1,\cdots,\mathbf v_m \}$ in $n$ dimensions, s.t. $m > n+1$ and a point $\mathbf c$ inside the convex hull, find an enclosing simplex of $\mathbf c$(of size $r \le n+1$). To solve the the problem, one can find a solution to $\alpha_1 \mathbf v_1+\cdots+\alpha_m \mathbf v_m=\mathbf c$ and $\alpha_1+\cdots+\alpha_m=1$. Once the solution is found, one can use the Carathéodory's theorem to reduce the number of non-zero ${\alpha_i}'s$ to $r$.

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