1. Lecture Note 1 – Linear Algebra Shuaiqiang Wang Department of CS & IS University of Jyväskylä http://users.jyu.fi/~swang/ shuaiqiang.wang@jyu.fi

2. Vectors Vector Facts: A vector is a point in a coordinate system! A vector has length and direction! (2,1) The length is The direction is

3. Subtract/Add Vectors (3,3) (1,2) (2,1)

4. Product of Vectors

5. Matrix Facts: A matrix is a collection of numbers! A matrix is a collection of vectors! A matrix is a mapping system of vectors!

6. Special Matrix

7. Transpose of Matrix

8. Subtract/Add Matrix

9. Product of Matrix

11. Inverse of Matrix

12. Mapping of Vectors Mapping SystemOriginal VectorMapped Vector How does it work? A matrix is a mapping system of vectors!

13. Simple Demonstration We use the 2-D system to demonstrate the mapping process Easy to calculate its eigenvalues and eigenvectors

14. Factorization of Square Matrix Square matrix Matrix composed of eigenvectors Diagonal matrix composed of eigenvalues Transpose of matrix For example:

15. Example Since Points (vectors) in the blue area are mapped into the red area! The directions of the mapping (in dashed lines) are decided by the directions of the eigenvectors! The strength of the mapping in two directions can be decided by the values of the eigenvalues!

16. Non-Square Matrix Square Matrix They share same eigenvalues! The vectors in P or Q are unit ones and orthogonal to each other!

17. Singular Value Decomposition (SVD)

18. Principal Component Analysis

19. We can keep the main mapping directions while remove some trivial directions:

20. Example Application in IR

21. SVD Application

22. Any Question?