Lecture Note 1 – Linear Algebra Shuaiqiang Wang Department of CS & IS University of Jyväskylä

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Presentation transcript:

Lecture Note 1 – Linear Algebra Shuaiqiang Wang Department of CS & IS University of Jyväskylä

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

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

Product of Vectors

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

Special Matrix

Transpose of Matrix

Subtract/Add Matrix

Product of Matrix

Inverse of Matrix

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

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

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

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!

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

Singular Value Decomposition (SVD)

Principal Component Analysis

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

Example Application in IR

SVD Application

Any Question?