SYSTEMATIC MATHEMATICS

Linear Algebra

This course contains 43 visible knowledge chapters across eight content-sized units of lengths 4, 5, 4, 6, 5, 6, 8, and 5. Suppose several equations describe the same unknown quantities. Is there one solution, no solution, or many—and how can you tell before doing pages of arithmetic? Begin by solving small systems and seeing each equation as a constraint. Rows and columns are compact records of those constraints; vectors describe combinations and directions; matrices describe linear rules. Then leave coordinate arrows long enough to see polynomials and functions as vectors, keep vectors distinct from dual measurements, define every norm used in a condition bound, and connect the four fundamental subspaces by orthogonal complements. The course proceeds through coordinate changes for both vectors and linear maps, determinants, orthogonal geometry, QR least squares, real and complex eigenpairs, Cayley–Hamilton recurrences, complex inner products, unitary and Hermitian maps, positive-definite quadratic forms, Cholesky, SVD, pseudoinverses, and low-rank approximation. Every symbol is read as an object or operation, every calculation is translated back into ordinary words, and the model capstone remains the final chapter after all required tools.

Before this course: Completed Algebra & Functions. Geometry helps interpretation but is not required; no calculus or programming is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A linear equation describes a flat constraint

Objective: Why does a doubled equation add no new information?

A linear equation combines unknowns only by scalar multiplication and addition, so its solution set is a line, plane, hyperplane, or an empty/degenerate case.

Start from the definition: A linear equation in x₁,…,xₙ has the form a₁x₁+⋯+aₙxₙ=b with fixed scalars aᵢ and b. The result to justify is: Two equations in two unknowns have one, none, or infinitely many common solutions; no fourth case exists.

The exact worked question is: Classify the intersection of x+y=5 and 2x+2y=10. Compare the result with this failure boundary: The equation 0x+0y=1 is not a line; it is a contradiction with no solution.

Two equations in two unknowns have one, none, or infinitely many common solutions; no fourth case exists.

Each nondegenerate equation defines a line in the coordinate plane.

Two distinct nonparallel lines meet once, parallel distinct lines never meet, and coincident lines share every point.

Degenerate equations reduce to either a universal truth or a contradiction, which fits the infinite or empty cases.

Classify the intersection of x+y=5 and 2x+2y=10.

  1. Divide the second equation by 2 to obtain x+y=5.
  2. Observe that both equations impose the same constraint.
  3. Parameterize with y=t and x=5−t for every real t.

Result: Infinitely many solutions: (x,y)=(5−t,t).