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Conditions for Linear transformation.

This chapter starts as follows rotations about the origin and all reflections in lines through the origin can be expressed as functions with rules of the form
x ---> Ax
where A is a 2 x 2 matrix
any function with such a rule is called a linear transformation

a linear transformation of the plane is a function of the form
f: R^2 --> R^2
x---> Ax
the transformation f is said to be represented by the matrix A

note eigenvectors have not been studied

there is some problems beginning

for each of the following functions f : R^2 ----> R^2
either explain why f is not linear , or write down the matrix that represents f

a) f(x,y) = (x+2y,y-x)
b) f(x,y) = (x+y+2,y-2x)
c) f(x,y) = (2,-1)
d) f reflects the plane in the line x = 2

with answers

a) (1 2)
-1 1

c) not linear transformation because it maps (0,0) to (2,1)

How do you get these and the other answers what is the reasoning behind these