# Please explain, this is a Linear transformation or not ?

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The second derivative of a four degree polynomial is a Linear Transformation or not?

The second derivative of a four degree polynomial is a Linear Transformation or not?

##### 1 Answer

See below

#### Explanation:

A trasformation

#T(v_1+v_2)=T(v_1)+T(v_2)# for every#v_1,v_2 \in V# #T(cv)=cT(v)# for every#v in V# and every scalar#c#

Note that the second property assumes that

When you derive a polynomial you lower its degree by **at most** four. In fact, a generic degree four polynomial is

If you want the degree two polynomial

With that being said, let's identify the polynomial space of degree

Let's proove the first property: assume we have the polynomials

and

This means that

(I applied twice the power rule for derivation: the second derivative of

Now let's compute

Similarly,

If you sum these expression, you can see that we have

The second property is shown in a similar fashion: given a polynomial

we have, for any real number

its second derivative is thus

which again is the same as computing