Linear Algebra with Sage        

Chapter 1. Vectors

*1.1 Vectors in n-space n-dim. space

    1.2  Inner product and Orthogonality

     1.3 Vector equations of lines and planes


Chapter 2. Linear system of equations

2.1 Linear system of equations

2.2 Gaussian elimination and Gauss-Jordan elimination


    Quiz :

Chapter 3. Matrix and Matrix Algebra


 3.1 Matrix operation

 3.2 Inverse matrix

 3.3 Elementary matrix

 3.4 Subsapce and linear independence

 3.5 Solution set of a linear system and matrix

 3.6 Special matrices

*3.7 LU-decomposition


Chapter 4. Determinant

4.1 Definition and Properties of the Determinants

4.2 Cofactor Expansion and Applications of the Determinants

4.3 Cramer's Rule

*4.4 Application of Determinant

4.5 Eigenvalues and Eigenvectors


*Chapter 5. Matrix Model

5.1 Lights out Game

5.2 Linear Model (Google)

5.3 Exam and Project


Chapter 6. Linear Transformations

6.1 Matrix as a Function (Transformation)

6.2 Geometric Meaning of Linear Transformations

6.3 Kernel and Range

6.4 Composition of Linear Transformations and Invertibility

6.5*Computer Graphics with Sage



Chapter 7. Dimension and Subspaces

7.1 Properties of bases and dimensions

7.2 Basic spaces of matrix

7.3 Rank-Nullity theorem

7.4 Rank theorem

7.5 Projection theorem

*7.6 Lleast square solution

7.7 Gram-Schmidt orthonomalization process

7.8 QR-Decomposition; Householder transformations

7.9 Coordinate vectors


Chapter 8. Diagonalization

8.1 Matrix Representation of LT

8.2 Similarity and Diagonalization

8.3 Diagonalization with orthogonal matrix, *Function of matrix

8.4 Quadratic forms

*8.5 Applications of Quadratic forms

8.6 SVD and generalized eigenvectors

8.7 Complex eigenvalues and eigenvectors

8.8 Hermitian, Unitary, Normal Matrices

*8.9 Linear system of differential equations



Chapter 9. General vector spaces

 9.1 Vector spaces

 9.2 Inner product spaces

 9.3 Isomorphism


Chapter 10. Jordan Canonical Form

10.1 Finding the Jordan Canonical Form with a Dot Diagram

*10.2 Jordan Canonical Form and Generalized Eigenvectors

10.3 Jordan Canonical Form and CAS



Final Exam:

   SKKU LA 2009- Final Exam


 Solutions :


Internet resources :

Sage-Reference: Link
Sage Tutorial:
2011-How to use Sage 2:
William Stein demos sage math:
2011-Mobile Math with Sage:
Sage Interact / ODE and Mandelbrot:
Sage Multivariable Calculus (1 of 2) by Travis:
Sage Multivariable Calculus (2 of 2) by Travis:
SOCW: Link (정의집)

Main Author : Sang-Gu Lee


Co-Authors : ***

Made by Prof. Sang-Gu LEE(sglee at

Copyright at Sungkyunkwan University, March 2013