Unit 2 of 11
Systems of Linear Equations
- 7 lessons
- Videos 6 to 12
A linear system asks for every input that satisfies several linear equations at once. This unit solves systems by elimination and describes all of their solutions.
Each system is read two ways, as intersecting lines or planes and as a target built from the columns of a matrix. Row operations reduce it to echelon form, and pivots and free variables describe every solution. A changing coefficient shows how a system can switch between one solution, none and infinitely many. A square matrix has an inverse exactly when each vector of the output space comes from exactly one input, and LU factorization stores elimination so it can be reused for new right-hand sides.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Linear Systems and Elimination
Linear Systems as Equations and Geometry
What a linear system and its solutions are, how to write a system in matrix form or as an augmented matrix, and how the row and column pictures show whether it is consistent or inconsistent.
Video 6
Gaussian Elimination and Row Operations
How elementary row operations simplify a system without changing its solutions, how elementary matrices record them, and how Gaussian elimination reaches row echelon form before back-substitution.
Video 7
Describing All Solutions
Pivots, Free Variables and All Solutions
How pivots and pivot columns split the unknowns into pivot and free variables, how an inconsistent row rules out solutions, and how reduced row echelon form gives a parametric solution, a particular plus a homogeneous solution.
Video 8
Linear Systems with Parameters
How to solve a family of linear systems that depends on a coefficient parameter, find the exceptional values where division fails, classify every case, and tell a parameter apart from a free variable.
Video 9
Inverses and LU Factorization
Inverse Matrices and Invertibility
What the inverse of a square matrix is, when a matrix is invertible or singular, how to compute an inverse by row reduction, why a pivot in every column decides invertibility of a square matrix, and how an inverse solves a system.
Video 10
LU Factorization and Pivoting
How Gaussian elimination stores its multipliers in a lower triangular L and its result in an upper triangular U, how forward and back-substitution reuse A = LU, and how permutation matrices and partial pivoting give PA = LU.
Video 11
Linear Systems Workshop
Linear Systems Workshop
How to classify a linear system before solving, spot redundant equations and inconsistent right-hand sides, write the complete solution family, handle an exceptional coefficient, track a row swap and check the original equations.
Video 12