Linear Algebra study guide, conventions and references

This graph covers an undergraduate linear algebra sequence, with abstract operator theory as a second pass.

## Learning path

1. [Systems and elimination](https://knowen.md/nodes/23411): represent a system, reduce an augmented matrix, identify consistency and free variables.
2. [Matrix operations](https://knowen.md/nodes/23460): check dimensions, multiply, transpose, conjugate, and distinguish inverses from adjoints.
3. [Spans and bases](https://knowen.md/nodes/23775): connect pivots to column bases, solve for coordinates, and use rank–nullity.
4. [Linear maps](https://knowen.md/nodes/23839): distinguish domain, codomain and range; represent maps in chosen bases.
5. [Orthogonality and least squares](https://knowen.md/nodes/23893): compute projections, QR and best fits.
6. [Eigenvalues](https://knowen.md/nodes/23680): compare algebraic/geometric multiplicity, diagonalize when possible, and understand spectral theorems.
7. [Factorizations and numerical methods](https://knowen.md/nodes/23622): choose LU, Cholesky, QR or SVD for the problem.
8. [Applications](https://knowen.md/nodes/24113): use consistent conventions for Markov chains, differential equations, coding and input–output models.

Study Jordan and rational canonical forms after the basis, map and eigenvalue material. They are not prerequisites for elementary least squares.

## Conventions and common traps

- Scalars are in a fixed field. Standard geometry uses $\mathbb R$ or $\mathbb C$; coding uses finite fields.
- Vectors are columns. A matrix with $m$ rows and $n$ columns maps $\mathbb F^n$ to $\mathbb F^m$.
- Over $\mathbb C$, $\langle x,y\rangle=x^*y$: the first argument is conjugated. Some textbooks use the opposite convention.
- Use the monic characteristic polynomial $p_A(t)=\det(tI-A)$.
- A basis is an ordered list when coordinates matter.
- Row operations preserve row span and null space, but generally change column space. Take pivot columns from the original matrix.
- Rank–nullity is $\operatorname{rank}A+\operatorname{nullity}A=n$, the **column count**.
- A characteristic polynomial splitting is insufficient for diagonalizability.
- A small determinant is not a reliable conditioning test.
- “Zero, one, or infinitely many solutions” assumes an infinite field. Over $\mathbb F_q$, a consistent rank-$r$ system has $q^{n-r}$ solutions.

## References for further study

These are reference resources, not quotations reproduced in this graph. Explanations and worked examples here are written for the graph.

- [MIT OpenCourseWare, 18.06SC Linear Algebra](https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/): lectures, exercises and solutions for a first course.
- [MIT 18.06 course materials](https://github.com/mitmath/1806): computational perspectives and course notes.
- [Sheldon Axler, Linear Algebra Done Right](https://linear.axler.net/index.html): a second-course treatment of vector spaces, operators and inner-product geometry.

Use the exercises in this graph to test understanding before reading their worked solutions.Copy