Generator of a fractional factorial design

To analyze which effects will be confounded in a fractional design, we shall introduce a new concept, the generator of a fractional factorial design.[1] As an example to illustrate the principles, we shall use a fractional factorial design To understand why the generators are practical to use we shall write down the complete variable matrix of a 2 factorial design. 

Confirm that these are the same experiments as were selected from the complete design. 

The matrix above has rather peculiar mathematical properties. If the columns are multiplied by each other, the result will always be another column in the matrix.  

Other examples of mathematical groups are the integer numbers on which multiplication and addidion is defined. Multiplication of two integer numbers will produce another integer number. The neutral element in multiplication is the number (1). Hie same holds for addition, where zero (0) is the neutral element. 

Multiplication by I does not change anything since we are multiplying the elements of the other colum by (+1). 

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