Applied Mathematics/The Basics

The Basics of linear algebra

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A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.

Basic operation[1]

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Operation Definition Example
Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise:
(A + B)i,j = Ai,j + Bi,j, where 1 ≤ im and 1 ≤ jn.

 

Scalar multiplication The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c:
(cA)i,j = c · Ai,j.
 
Transpose The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa:
(AT)i,j = Aj,i.
 

Practice problems

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(1)  
(2)  
(3)  

Matrix multiplication

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Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B[2]

 [3]


 
Schematic depiction of the matrix product AB of two matrices A and B.



Example

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Practice Problems

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(1)  

(2)  


Dot product

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A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.

Suppose A is row vector and B is column vector, then the dot product is defined as follows;

 

or

 


Suppose   and   The dot product is

 

Example

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Suppose   and  

 

 
 

 

Practice problems

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(1)   and  

 

(2)   and  

 

Cross product

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Cross product is defined as follows:

 

Or, using detriment,

 

where   is unit vector.

References

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  1. Sourced from Matrix (mathematics), Wikipedia, 28th March 2013.
  2. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.
  3. Sourced from Matrix (mathematics), Wikipedia, 30th March 2013.