# Span of a subspace, with Polynomials

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 $K$?

• 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. – Lubin Sep 6 '17 at 21:29
• you are correct the polynomials of degree two – Temirzhan Sep 6 '17 at 21:32
• 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? – lulu Sep 6 '17 at 21:40
• 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 – 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.
• @user203042, if you get a row-reduced matrix of rank $3$ then the vectors span the whole space. – lhf Sep 6 '17 at 22:04
You can express the polynomials $1,t$ and $t^2$ by linear combinations of the given polynomials. $\ (1)$
We can remove an item from our list till as long the remaining ones satisfy $(1)$.