Machine Learning From scratch | Part 3. Matrices and matrices Dot Product

Prerequisit  : Vector Dot Products

Matrices are grid of numbers or arrangment of numbers . In programming it can be an array of array

a real example can be  marks of students (A,B,C) in subjects (S1,S2,S3)

can be shown as Student

A have scored  20 in S1 , 25 in S2 and 30 in S3  = [20,25,30]

B have scored  25 in S1 , 15 in S2 and 20 in S3  = [25,15,20]

C have scored  35 in S1 , 25 in S2 and 30 in S3  = [35,25,30]

   		 S1    S2    S3     ----> Row
          
 	A 	 20    25    30    
 
 	B 	 25    15    20
 
 	C 	 35    25    30
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    V
    Columns

Matrix Shape

Every matrix has a shape which is  

number of rows * number of columns 

for our above matrix we can say it is a 3*3 matrix as it has 3 rows and 3 columns

Matrix Dot Products

we have one matrix A of shape 2*3 (which means we have 2 row and 3 column ) and we have one more matrix B of shape 3*2 (which means we have 3 rows and 2 column   )

for matrix multiplications thier is one rule which should be statisfied
A(2*3) B(3*2) the number of columns of first matrix and number of rows in second matrix showld be equal

in First matrix no columns is 3
in Second matrix no rows is   3

the result will be a matrix of shape numer of rows from first matrix * number of columns in second matrix for our example 

result = matrix of shape 2*2 (2 rows and 2 columns)


wo we can multiply this two matrices 

A =   a1  a2  a3     B = b1  b2
      a4  a5  a6         b3  b4
                         b5  b6
                      
AB =   a1b1 + a2b3 + a3b5         a1b2 + a2b4 + a3b6
       a4b1 + a5b3 + a6b5         a4b2 + a5b4 + a6b6  


Lets See by Example 


A = [[1,2,3],[4,5,6]]
B   = [[1,2],[3,5],[4,2]] --> this has 3 rows and 2 column


A = 1  2  3         B  = 1  2
    4  5  6              3  5
                         4  2 
AB = 1*1 + 2*3 + 3*4           1*2 + 2*5 + 3*2
     4*1 + 5*3 + 6*4           4*2 + 5*5 + 6*2
   
   = 1+6+12    2+10+6     
     4+15+24   8+25+12      
     
   = 19 18  
     43 45
   

In the next article we will see how to calculate it in program