Vectors on

• ordered tuple:
  
  

• column vector:
  
  

• row vector:

Transpose

• swap rows and columns
  
  

• example:
  
  

• apparently

Addition

Scalar multiplication

Vector space

• a set with specific addition and scalar multiplication rules
  
  

• a vector is an element in the vector space
  
  

• example: polynomials
  
  

• example: continuous functions on
  
  

Linear dependence

• Given
  
  

• if there exist , with at least one of which being non-zero, so that ，then are said to be linearly dependent
  
  

• otherwise if holds only at ，then they are called linearly independent

Example

• and are linearly independent
  
  

• 
  
  

Example

• 、、 are linearly dependent
  
  

Motivation

If , and ,

then can be written as linear combinations of other vectors:

Exercise

• Let
  
  
  

• are linearly dependent or independent？
  
  

• how about ？

Exercise

• Determine linear dependence of the two sets below respectively
  
  

• 
  
  

• 
  
  

Tips

• Let be a set of vectors
  
  

• If ，then vectors in are linearly dependent
  
  

• If ，and ，then vectors in are linearly dependent
  
  

Exercise

• Let
  
  
  

• write into linear combinations of

Summary

• Concepts
  
  
• Addition and scalar multiplication
  
  
• Linear Dependence