NO.Idea

Linear Algebra

What is a system?

A system has input and output. Hence, system is also called function, transformation, operator.

Ex: Speech Recognition System, Dialogue System, Communication System.

Linear System#

properties:

  1. Preserving Multiplication
  1. Preserving Addition

Non-linear System

Terminology#

one-to-one: no two inputs has same output

onto: co-domain = range

Properties of Vector#

  • u+v=v+u\vec{\bm{u}} + \vec{\bm{v}} = \vec{\bm{v}} + \vec{\bm{u}}
  • (u+v)+w=u+(v+w)(\vec{\bm{u}} + \vec{\bm{v}}) + \vec{\bm{w}} = \vec{\bm{u}} + (\vec{\bm{v}} + \vec{\bm{w}})
  • 0+u=u\vec{\bm{0}} + \vec{\bm{u}} = \vec{\bm{u}}
  • u+u=0\vec{\bm{u}}^{'} + \vec{\bm{u}} = \vec{\bm{0}}
  • 1u=u1\vec{\bm{u}} = \vec{\bm{u}}
  • (ab)u=a(bu)(ab)\vec{\bm{u}} = a(b\vec{\bm{u}})
  • a(u+v)=au+ava(\vec{\bm{u}} + \vec{\bm{v}}) = a\vec{\bm{u}} + a\vec{\bm{v}}
  • (a+b)u=au+bu(a+b)\vec{\bm{u}} = a\vec{\bm{u}} + b\vec{\bm{u}}

Matrix#

(AT)T=A(A^T)^T=A

Readings 🔗#

  • Linear Algebra | Mathematics | MIT OpenCourseWare