Linear Algebra - Linear Operators

  • #1

Homework Statement



True or false?
If T: ℙ8(ℝ) → ℙ8(ℝ) is defined by T(p) = p', so exists a basis of ℙ8(ℝ) such that the matrix of T in relation to this basis is inversible.

Homework Equations




The Attempt at a Solution



So i think that my equations is of the form:
A.x = x'
hence A is non-singular and therefore is inversible. But my exercise anwser say the opposite.
Could somebody elucide me?

P.s: (sorry for bad english, I'm from Brazil and still learning it.)
 
Last edited:

Answers and Replies

  • #2
35,052
6,791

Homework Statement



True or false?
If T: ℙs → ℙs is defined by T(p) = p', so exists a basis of ℙs such that the matrix of T in relation to this basis is inversible.

Homework Equations




The Attempt at a Solution



So i think that my equations is of the form:
A.x = x'
hence A is non-singular and therefore is inversible.
You need to give a reason that A is nonsingular, if in fact this is a true statement.

BTW, we say "invertible" if an inverse exists.
Victor Feitosa said:
But my exercise anwser say the opposite.
Could somebody elucide me?
What does this transformation do to the standard basis for Ps? Also, what is Ps? Was this supposed to be P5, the space of polynomials of degree < 5?
Victor Feitosa said:
P.s: (sorry for bad english, I'm from Brazil and still learning it.)
 
  • Like
Likes Victor Feitosa
  • #3
Hello, my friend! Thanks for your feedback.

I thought that A is non-singular because if it was, A.x = x' could not be true, because it would have infinite solutions. But thinking right now i don't see how this is true at all.
No, it's not P5, it's P8. I thought it was a generic polynomial space.
P8 is the space of real polynomials of degree < 8.
I'll edit first post.
 
  • #4
I see why this is false!!!!

My matrix A is a echeloned matrix with trace 0. So it's det is 0 and it's not invertible.
Sorry for not posting how i found this. I will try to edit and post my anwser!

Thanks PF.
 
  • #5
HallsofIvy
Science Advisor
Homework Helper
41,833
964
Hello, my friend! Thanks for your feedback.

I thought that A is non-singular because if it was, A.x = x' could not be true, because it would have infinite solutions.
You seem to have "singular" and "non-singular" reversed! This linear operator is singular precisely because it is not one- to- one.
If the difference between polynomials p and q is a constant, then Ap= p'= q'= Aq so A inverse of p' is not unique.

But thinking right now i don't see how this is true at all.
No, it's not P5, it's P8. I thought it was a generic polynomial space.
P8 is the space of real polynomials of degree < 8.
I'll edit first post.
 
  • Like
Likes Victor Feitosa
  • #6
You seem to have "singular" and "non-singular" reversed! This linear operator is singular precisely because it is not one- to- one.
If the difference between polynomials p and q is a constant, then Ap= p'= q'= Aq so A inverse of p' is not unique.

Oooooh, so logic and true! Thank you.
So, a matrix is singular if it's determinant is zero hence it have no inverse, is it right?
 
  • #7
35,052
6,791
Oooooh, so logic and true! Thank you.
So, a matrix is singular if it's determinant is zero hence it have no inverse, is it right?
Yes.
Regarding the problem in your first post, you need to show that if B is any basis for P8, the matrix for the transformation is not invertible. What you've shown is the matrix in terms of the standard basis is noninvertible.
 
  • Like
Likes Victor Feitosa
  • #8
Thanks everyone who helped! I think i figured how to do it and will try soon and post my result!
 

Related Threads on Linear Algebra - Linear Operators

  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
12
Views
3K
  • Last Post
Replies
0
Views
2K
Replies
1
Views
1K
  • Last Post
Replies
16
Views
6K
  • Last Post
Replies
3
Views
2K
Replies
5
Views
3K
Replies
2
Views
613
Replies
1
Views
5K
Replies
7
Views
6K
Top