Thread: Linear transformation proof

Hi,

I am trying to prove the following are not a linear transformation:

1. defined by

2. , where P is the vector space of polynomials (of any degree)

3.


My attempt:

I have tried the 0 vector & -1 scaler multiplication and it staisfies all the 3 eqations. Furthermore they satisfy addition as well. So why are they not linear transformation?

Originally Posted by M.R

Hi,

I am trying to prove the following are not a linear transformation:

1. defined by

2. , where P is the vector space of polynomials (of any degree)

3.


My attempt:

I have tried the 0 vector & -1 scaler multiplication and it staisfies all the 3 eqations. Furthermore they satisfy addition as well. So why are they not linear transformation?

The first one is not linear because it does not satisfy



Let and

This gives

but



For number 2. It is most definitely a linear operator. Differentiation is one of the first linear operators people learn.

For number 3. Try addition again

Note that



use the same idea as in number 1