Linear Systems Summary

Linear Algebra (Matrices) · Axiom Academy

SUMMARY Solving Linear Systems Everything from the unit in one place: augmented matrices, the three row operations, Gaussian and Gauss-Jordan elimination, and reading off every type of solution. A linear system is encoded as an augmented matrix — one row per equation, one column per variable, the bar separating coefficients from constants. The three elementary row operations — swap, scale by a nonzero constant, and add a multiple of one row to another — never change the solution set. Gaussian elimination drives the matrix to row-echelon form (a forward pass), then back-substitution reads the answer from the bottom up. Gauss-Jordan goes further to reduced row-echelon form (RREF), where the solution is read off directly. The RREF of a matrix is unique . The number of pivots is the rank , and rank decides everything: a system is consistent, unique, or has free variables based on it. Core Concept Augmented Matrices A system of equations becomes a single matrix : each equation is a row, each variable a column, and the vertical bar holds the constants apart from the coefficients. Why it helps: matrix row operations are equation manipulations — the bookkeeping disappears. Watch out for: keep variables in the same column order in every equation before you build it. Core Concept Elementary Row Operations Three reversible moves: swap two rows, scale a row by a nonzero constant, or replace a row with itself plus a multiple of another. Every one preserves the solution set.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.