Consider the system of linear equations for the two unknowns and ,
where are constant numbers. This system can be easily solved: solve the first equation for ,
and insert it into the second equation,
For this general solution for and to be valid, the denominator apparently has to be different from zero.
We write the two unknowns and as the components of a two–dimensional vector ,
Then, we write the two constants and as the components of a two–dimensional vector
The two–by–two system of linear equations, Eq. ( 5.1 ), maps the vector onto the vector . We write this in the following abstract form:
where we defined the two–by–two matrix
A two–by–two matrix is a quadratic scheme which, upon operating on a vector on its right, transforms this vector into another vector according to the rule
By comparison we recognise that this matrix equation, , is equivalent to the system Eq.( 5.1 ).
A linear mapping A from maps a vector onto the vector . The mapping is represented by a two-by-two matrix . The mapping must fulfill
We compare this to