Math Is Fun Forum

freddogtgj

Member

Registered: 2006-12-02

Posts: 54

Linear Algebra

Question that is posing a problem is:

Determine which of the following are linear transformations from P_2 to P_2:

(a) (L(p))(t)=t+p(t)

(b) (L(p))(t)=tp'(t)+p''(t)

Thanks in advance!

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Linear Algebra

Your notation would throw me off if I hadn't known what you were talking about.  May I suggest writing the functions this way:

L(p(t))

It seems to make it a bit easier on the mind.  Not saying your way is wrong, and if it's what your book/professor uses, then by all means use it.  I just don't like it.

As for your question, linear transformation problems just involve plugging in a sum of two general element from the domain, finding out what it is in the range, and then seeing if it's a sum of the two general elements.  The same goes for multiplication.  So in your first example:

L(f + g) = t + f + g
L(f) = t + f
L(g) = t + g

That should be enough for you to see that in general, L(f+g) is not equal to L(f) + L(g).  Now we try the same for the 2nd function:

L(f + g) = t*(f+g)' + (f+g)'' = t*f' + t*g' + f'' + g''

Now it's up to you to finish it.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."