Difference between linear and nonlinear transformation

[yaganon]

right now, my concept for their difference is that linear transformations are 1 to 1, where as nonlinear transformations are not. However, P^n to P^(-1) is a linear transformation, but it's not 1 to 1.

the textbook def of linear transformation is that it must be closed under addition and scale factorization.

I'm confused.

 

[Office_Shredder]

A linear transformation is just one that satisfies the conditions of linearity:

T(x+y)=T(x)+T(y)
T(ax)=aT(x)

where x and y are vectors, a a scalar. Nothing requires linear transformations to be 1 to 1 (they usually aren't) or nonlinear transformations to not be 1-1

 

[HallsofIvy]

right now, my concept for their difference is that linear transformations are 1 to 1, where as nonlinear transformations are not. However, P^n to P^(-1) is a linear transformation, but it's not 1 to 1.

the textbook def of linear transformation is that it must be closed under addition and scale factorization.

I'm confused.

Yes, you are. It makes no sense to talk about a transformation as being "closed under addition and scale factorization". Only sets are "closed" under given operations, not transformations. A "linear transformation" on a vector space is a function, F(v), such that F(au+ bv)= aF(u)+ bF(v). They are certainly NOT, in general, "one-to-one", as Office Shredder says. For example, any projection is linear but is not one-to-one.