linear-algebra

Linear algebra — working toolkit

The companion to applied-math. That skill tells you what to look for in a system; this one tells you how to compute it once the system turns out to be linear. Use it as a reference to dip into, not a course to read through.

Diagnostic first — what kind of question is this?

Question	Section
"Solve Ax = b"	Systems
"How many solutions, and why?"	Systems → Rank
"Is this transformation reversible?"	Inverses, Determinants
"How many independent things are really here?"	Rank, Basis and dimension
"What is conserved / what costs nothing?"	Four fundamental subspaces
"No exact solution exists — what's closest?"	Orthogonality, Least squares
"What are this system's natural modes / long-run behavior?"	Eigenvalues, Diagonalization
"This is numerically unstable"	Conditioning

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1. Systems of equations

A linear equation has unknowns only in numerators, only to the first power, with no products or quotients of unknowns: a₁x₁ + a₂x₂ + ⋯ + aₙxₙ = b. Anything else is not this toolkit's problem until you linearize it.

The augmented matrix [A | b] strips the equations to their coefficients. Three elementary row operations — swap two rows, scale a row by a nonzero constant, add a multiple of one row to another — change the representation without changing the solution set. That invariance is the whole method.

Gaussian elimination reduces to row echelon form (leading 1 in each nonzero row, each further right than the one above, zero rows at the bottom), then back-substitutes. Gauss-Jordan continues to reduced row echelon form (rref — additionally, each leading 1 is the only nonzero entry in its column), reading the solution off directly. Gauss-Jordan does more arithmetic; Gaussian elimination plus back-substitution is usually cheaper. rref is canonical: a matrix has exactly one.

Reading the outcome. Exactly three possibilities, always:

• No solution — a row reads [0 0 ⋯ 0 | c] with c ≠ 0, asserting 0 = c. Inconsistent.
• Exactly one solution — every column has a pivot; no free variables.
• Infinitely many — consistent with at least one pivot-free column. Each pivot-free column is a free variable; parameterize it (x₃ = t) and express the pivots in terms of the free ones. The number of free variables is the dimension of the solution family, and that family's structure is often the actual result, not a failure to pin one down.