Linear combinations as maps

• fix
  
  

• given , form the linear combination
  
  

• viewing coefficients as an vector，this induces an map:

Example

• fix
  
  

• for any , there is the linear combination
  
  

• which produces the map:

Matrix

• fix ，writing these column vectors from left to right in order, we obtain an (real) matrix
  
  

• the set of all real matrices if often denoted by
  
  

• Typically we use capital letters (e.g., ) for matrices, and for the entry at the intersection of the -th (from top down) row and -th (from left to right) column in
  
  

Multiplying a vector

• given an matrix with columns
  
  

• let
  
  

• we define the multiplication to be the linear combination

Exercise

• Compute the following multiplications:
  
  

Transforming linear systems

• There are some some rabbits and chicken in the barn. What will be their respective numbers if there are 94 legs and 35 heads in total?
  
  

• 
  
  

• rewrite as

Transpose

• given an matrix with columns
  
  

• swapping its rows and columns, we obtain a new matrix with rows
  
  

• this new matrix is denoted by , and apparently

Multiplication from left

• given an matrix ，and and row vector
  
  

• define to be
  
  

• again the dimension shall align

Exercise

• Compute the following multiplications:
  
  

Symmetric matrices

• a matrix is called symmetric if
  
  

• apparently symmetric matrices are square matrices (i.e., with equal number of rows and columns)
  
  

• given an square matrix , the line from top left to bottom right consisting is called the main diagonal of
  
  

• symmetric entries are symmetric w.r.t the main diagonal

Examples and non-examples

• is symmetric, while is not
  
  

• The main diagonal of is

Exercise

find , at least one of which is non-zero and , again at least one of which is non-zero, so that the following holds

Exercise

• Are the following two maps injective? In general, when is injective？
  
  

• 
  
  

Summary

• Matrices and notations
  
  

• Multiplying with column vectors and its implications
  
  

• Matrix transpose and symmetricity