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M02.6 · Linear systems and elimination

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: A system is a question about a map; Elimination, step by step; None, one, or infinitely many; Elimination as matrices.

M02.6 L1 · Core 4 lessons · about 110 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

A matrix turns an input into an output. A linear system runs that machine backwards: it gives you the output and asks for the input. These four lessons read a system as lines that meet and as columns that combine, solve it by Gaussian elimination one justified step at a time, explain why the answer is always none, one or infinitely many, and show that elimination is itself a product of matrices, which is how numerical libraries solve systems.

What you need first. Linear combinations and span (M02.2) and matrices as linear maps, including the column view and the row view of a matrix times a vector (M02.4). The last lesson also uses matrix multiplication and the identity matrix (M02.5).

Lessons

1. A system is a question about a map M02.6.1 25 min

   A linear system Ax = b hands you the output of a linear map and asks for the input. See the same two equations as lines that cross and as columns that add up, and learn how to tell whether a target can be reached at all.

2. Elimination, step by step M02.6.2 30 min

   Gaussian elimination solves a linear system with three kinds of row operation. Work through a three-by-three system by hand, see why each operation keeps the solutions exactly as they were, and finish with back substitution.

3. None, one, or infinitely many M02.6.3 25 min

   A linear system has no solution, exactly one, or infinitely many, and elimination tells you which. Learn to read the three cases from the last rows of the matrix, describe a whole line of solutions with a free variable, and prove that exactly two solutions can never happen.

4. Elimination as matrices M02.6.4 30 min

   Every row operation is multiplication by a matrix, and elimination as a whole is the factorisation A = LU. See why the multipliers drop straight into L, solve with two triangular systems, count the work, and learn why numerical code calls solve and does not form an inverse.

The interactive

M02.6-X1 · Elimination stepper. Apply row operations to an augmented matrix with buttons and watch the lines or planes turn while the point they share stays exactly where it was. It lives in Elimination, step by step.