I need some help, I haven't learned anything in Linear my professor's teaching style just doesn't work for me. So i have no examples to work off of on how to do this problem.

I have the following polynomials

  1. $p_1(t)=1+t$
  2. $p_2(t)=1-t$
  3. $p_3(t)=4$
  4. $p_4(t)=t+t^2$
  5. $p_5(t)=1+2t+t^2$

Let $K = \text{Span}\{p_1,p_2,p_3,p_4,p_5\}$ a sub-space of $P^2$

How can I determine what polynomials can be removed without changing $K$?

  • $\begingroup$ 1. I’ve edited to put your post into proper MathJax, and hope that I have not misinterpreted your intent. In particular, I don’t know whether that should have been $P_2$ instead of $P^2$. And I presume that “P2” meant the polynomials of degree at most two? 2. You should learn basic TeX/MathJax if you intend to post again. $\endgroup$ – Lubin Sep 6 '17 at 21:29
  • $\begingroup$ you are correct the polynomials of degree two $\endgroup$ – Temirzhan Sep 6 '17 at 21:32
  • 1
    $\begingroup$ Hint: a good start would be to determine the dimension of $K$. It is clearly at least $1$ and at most $5$. Can you compute it exactly? $\endgroup$ – lulu Sep 6 '17 at 21:40
  • $\begingroup$ Hmmm I think I get it so If i set the Polynomials up in a matrix and row reduce I have 3 pivot columns so the dim(K)=3 $\endgroup$ – Temirzhan Sep 6 '17 at 21:56

Hint: Write the given polynomials as vectors with respect to the basis $1,t,t^2$. Put them as rows in a $5 \times 3$ matrix and use row reduction.

  • $\begingroup$ Ok so after doing that and row reducing I see that I can get rid of P3(t)=4 && P5(t) = 1+2t+t^2 right? $\endgroup$ – Temirzhan Sep 6 '17 at 21:58
  • $\begingroup$ @user203042, that's one solution, yes. $\endgroup$ – lhf Sep 6 '17 at 22:01
  • $\begingroup$ Okay so i'm a bit confused still If I remove those two polynomials because I still have a set of Linear independent vectors that span P3? and therefore span the subspace? $\endgroup$ – Temirzhan Sep 6 '17 at 22:02
  • $\begingroup$ @user203042, if you get a row-reduced matrix of rank $3$ then the vectors span the whole space. $\endgroup$ – lhf Sep 6 '17 at 22:04
  • $\begingroup$ Ah okay, we haven't learned about rank yet unfortunately.. So i'm trying to say it in terms that would make sense for the beggining of the course $\endgroup$ – Temirzhan Sep 6 '17 at 22:11

You can express the polynomials $1,t$ and $t^2$ by linear combinations of the given polynomials. $\ (1)$
(Therefore, you can express all at most quadratic polynomials.)
We can remove an item from our list till as long the remaining ones satisfy $(1)$.


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