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Conditions for Linear transformation.

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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

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