Linear Algebra Notebook

Linear algebra notebook

Linear Algebra Notebook

中文

We use a notebook for doing computation, finding examples, or taking notes. The idea of this Linear Algebra Notebook (LA-notebook) follows the same philosophy: Whenever you encounter a new mathematical object, you should play with it by doing some computation , run some examples to see its behaviors (both examples or counterexamples!), and then write down the notes just for yourself . This process is like conducting experiments in your mind, where your brain is analyzing the mathematical objects.

Take a awesome note for yourself, so that you can easily review what your brain has done in the future!

To download the LA-notebook：Visit LA-notebook Releases and download the latest version; newer versions are at the top, and you may click on “Source code (zip)” to download it.

Usage

You need Jupyter with the SageMath kernel to open the files.

You may choose one of the following two methods.

1. Install SageMath on your local machine (faster)

  1. Follow the Installation Guide to install SageMath
  2. Download the LA-notebook through the link at the top of this page
  3. Use the Jupyter Notebook, which was installed by SageMath, to open the ipynb files in the LA-notebook.

If you are not able to install SageMath on your machine, or you wish to use the browser version of it. You may run the files through CoCalc as follows.

2. Run the files through CoCalc (slower)

  1. Register a CoCalc account
  2. Download the LA-notebook through the link at the top of this page
  3. Login and then click on “Create New Project…” (you may enter any project name you like)
  4. Click on “Upload” and select LA-notebook.zip you downloaded in Step 2.
  5. Click on the zip to extract it.

Table of contents

(The links below are read-only. Follow the instructions above if you wish to run the code inside a file.)

1. Linear geometry

  1. Vector, length, and angle
  2. Subspaces in Rn
  3. Column space of a matrix
  4. Row space of a matrix
  5. Projection and reflection
  6. Affine subspaces in Rn
  7. Solution set of Ax = b
  8. Row operations
  9. Finding a particular solution
  10. Finding the homogeneous solutions
  11. Number of solutions
  12. Matrix inverse
  13. Elementary matrices
  14. Four fundamental subspaces

a. Sage: Matrices and linear equations

Basic geometry & subspaces
101 --> 102 --> {103, 104} --> 105

Affine subspace & solutions
(102 -->)
106 --> 107 --> 108 --> {109, 110} --> 111

Topics
112, 113, 114, 10a

2. Linear spaces

  1. Linear independence
  2. Basis
  3. Column space, left kernel, and their bases
  4. Row space, kernel, and their bases
  5. Basis exchange lemma
  6. Dimension, expanding and shrinking lemmas
  7. Rank and nullity
  8. Vector space
  9. Subspaces in a vector space
  10. Common vector spaces
  11. Constructing new subspaces
  12. Constructing new vector spaces
  13. Orthogonal geometry
  14. Gram–Schmidt orthogonalization
  15. Direct sum of orthogonal subspaces
Spaces in Rn
201 --> 202 --> {203, 204} --> 205 --> 206 --> 207

Abstract spaces
208 --> 209 --> 210

Operations of spaces
211 -->  212

Inner product space
213 --> 214 --> 215

3. Linear functions

  1. Function basics
  2. Linear function
  3. Matrix as a linear function
  4. Linear function as a matrix
  5. Vector representation in Rn
  6. Vector representation in a vector space
  7. Change of basis
  8. Isomorphism
  9. Matrix representation in Rn
  10. Matrix representation in a vector space
  11. Lagrange polynomials and Vandermonde matrix
  12. Sylvester matrix and resultant
  13. Understanding the spectral decomposition
  14. Understanding the singular value decomposition
  15. Understanding the Jordan canonical form
Linear function
301 --> 302 --> 303 -->304

Vector and matrix representations
305 --> 306 --> 307 --> 308 --> 309 --> 310

Topics
311, 312, 313, 314, 315

4. Determinant

  1. Determinant for small matrices
  2. Invertible matrix as the product of elementary matrices
  3. Elementary matrix acts on columns
  4. Elementary matrix acts on rows
  5. Definition of the determinant
  6. Invertibility
  7. Matrix multiplication and transpose
  8. Block matrix
  9. Distributive law and Laplace expansion
  10. Adjugate
  11. Cramer’s rule
  12. Permutation matrix
  13. Permutation expansion
  14. Interpretation through graph theory
  15. Determinant is well-defined
Determinant for small matrices
401

Geometric interpretations
402 --> {403, 404}

Definition and properties
405 --> {406, 407, 408} --> 409

Adjugate
410 --> 411

Permutation expansion
412 --> 413 --> {414, 415}

5. Diagonalization

  1. Find a good basis
  2. Quadratic curve
  3. Recurrence relation
  4. Linear differential equation
  5. Matrix exponential
  6. Characteristic polynomial
  7. Coefficients of the characteristic polynomial
  8. Diagonalization
  9. Algebraic multiplicity and geometric multiplicity
  10. Eigenspace
  11. Graph and characteristic polynomial
  12. Cayley–Hamilton theorem
  13. Minimal polynomial
  14. Jordan canonical form

a. Python: NumPy and numerical linear algebra

Good basis
501 --> {502, 503, 504, 505}

Characteristic polynomial
506 --> 507

Diagonalization
508 --> 509 --> 510

Topics
511, 512, 513, 514, 50a

6. Theory of symmetric matrices

  1. Reduction
  2. Schur triangulation
  3. Symmetric matrices and normal matrices
  4. Spectral decomposition
  5. Singular value decomposition
  6. Understanding the principal component analysis
  7. Inertia
  8. Positivity
  9. Rayleigh quotient
  10. Cauchy interlacing theorem
  11. Equitable partition
  12. Covariance matrix
  13. Principal component analysis
  14. Laplacian matrix
  15. Spectral embedding and clustering
Schur triangulation
601 --> 602 --> 603 --> 604 --> 605 --> 606

Symmetric matrix
607 --> 608 --> 609 --> 610 --> 611

Statistics
612 --> 613

Laplacian matrix
614 --> 615

Appendix

A. Index with translations