Determine if a transformation is linear

[aero_zeppelin]

Well, I've done some research on the further, more advanced chapters in my course and learned that these two conditions must be met:

T(u + v) = T(u) + T(v)
T(ku) = kT(u)


I was wondering if there's another way to figure if the transformation is linear (since this question is being asked in one of the "basic" chapters previous to the one mentioning the formulas)

Thanks!

 

[aero_zeppelin]

btw... What would be a good definition for LINEAR TRANSFORMATION?

 

[BruceW]

search linear map on wikipedia. That gives pretty much the same definition as you have, but they say it a bit more rigorously.

 

[HallsofIvy]

Well, I've done some research on the further, more advanced chapters in my course and learned that these two conditions must be met:

T(u + v) = T(u) + T(v)
T(ku) = kT(u)


I was wondering if there's another way to figure if the transformation is linear (since this question is being asked in one of the "basic" chapters previous to the one mentioning the formulas)

Thanks!

The point of what aerozeppelin and Bruce W are saying is that "T(u+v)= T(u)+ T(v)" and "T(ku)= kT(u)" are usually taken as the definition of "linear transformation". And you certainly can't ask whether something is a linear transformation before you have defined it! If your text is asking to determine whether a function is a linear transformation before those formulas are given, what definition is given?

 

[aero_zeppelin]

The first time you get the LINEAR TRANSFORMATION idea in my text, they say:

" In the special case where the equations in 1 are linear, the transformation
T: Rn --> Rm defined by those equations is called a linear transformation (or a linear operator if m = n ). "And then, the question asked is:
" Show that the orthogonal projections on the coordinate axes are linear operators, and find their standard matrices"

How can this be shown?