1.3 Systems, rank, inverse, determinant
You can tell when a system has no solution, one, or infinitely many — and what that means for a model.
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Rank and invertibility decide whether a system of equations — and by extension a linear model — has one answer, none, or infinitely many. This topic covers elimination, the fundamental subspaces, and ill-conditioning, which explains why some regressions produce coefficients that flip sign on tiny data changes. It sits mid-module as the bridge from geometry to computation. The confusion worth naming: a solver that returns wild numbers has not necessarily failed — the problem itself may be ill-posed.
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Gaussian elimination and row reduction
The systematic procedure for solving a system of linear equations by simplifying it one row at a time. It is mechanical, and doing a few by hand is what makes the ideas that follow concrete.
Rank, null space, column space
How many genuinely independent directions a matrix contains, what it sends to zero, and what it can produce. These three answer whether a system has no solution, one, or infinitely many.
Invertibility, pseudo-inverse, ill-conditioning
Whether a transformation can be undone, what to do when it cannot, and the warning sign that a computed answer will be unreliable. Ill-conditioning is the practical reason numerical results sometimes disagree with theory.
Determinant as a volume scale factor
The determinant tells you how a transformation scales volume, and a determinant of zero means the transformation flattens space. That geometric reading makes the algebraic rules far easier to remember.
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