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Give an example of a $3$-dimensional subspace of $P_4$ which contains the polynomials $3$+$2t^2$, $t^4$, and $1+2t+3t^3$.

I have no idea how to even get started with this problem. Can I get some help? (Studying)

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If the three polynomials $p_1(t)= 3+2 t^2$, $p_2(t) = t^4$ and $p_3(t) = 1 + 2t + 3 t^3$ are linearly independent, then the can be used as a basis to form a 3 dimensional subspace.

To show that they are linearly independent, I need to show that if $\sum_{k=1}^3 \alpha_k p_k = 0$, then the $\alpha_k = 0$. This gives $\sum_{k=1}^3 \alpha_k p_k(t) = (3 \alpha_1+\alpha_2)+2 \alpha_2 t + 2 \alpha_1 t^2+3 \alpha_2 t^3 + \alpha_3 t^4 = 0$. If we differentiate 4 times, we get that $\alpha_3 = 0$. If we differentiate 3 times and set $t=0$, we get that $\alpha_2 = 0$. Just setting $t=0$ gives $\alpha_1=0$, hence these three polynomials are linearly independent.

So, a three dimensional space that contains the given polynomials is given by $S = \text{sp} \{p_k\}_{k=1}^3 = \{ \sum_{k=1}^3 \alpha_k p_k \}_{\alpha \in \mathbb{R}^3}$, where $p_1,p_2,p_3$ are given by above.

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Just curious if I'm missing the obvious, but where does the + 4t^4 come from? Did you combine the first two polynomials? –  LearningPython Nov 7 '12 at 5:41
    
My apologies, I misread the question, I thought there were two polynomials. I will fix the response. –  copper.hat Nov 7 '12 at 5:50
    
OK, I fixed it, sorry about that! –  copper.hat Nov 7 '12 at 5:58
    
No prob! Just making sure. I'm reading your solution now. Thank you so much for your insight. –  LearningPython Nov 7 '12 at 6:03
    
This answer was fantastic, albeit a bit over my head. I believe this is an Einstein Summation? I'm assuming part of what it does can be labeled synonymous with the definition of linear independent if it = 0. I will study until I understand. –  LearningPython Nov 7 '12 at 6:22
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Perhaps $\text{span}\{ 3+2t^2,t^4,1+2t+3t^3 \}$? I suppose you need to argue that the set in question is indeed linearly independent, but that shouldn't be hard.

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What exactly is P_4? –  LearningPython Nov 7 '12 at 4:38
    
I'd expect that someone asking a question here would at least understand the symbols he's using...anyway, $\,P_4\,$ usually denotes the set of all the polynomials of degree less than or equal $\,4\,$, over some fixed and given field (usually, the reals) –  DonAntonio Nov 7 '12 at 4:44
    
Ah, that's pretty straight forward. It was unclear to me. Is the fact that the subspace must be 3-dimensional significant to the answer, or would I just span the polynomials and/or prove that they are linearly (in)dependent? –  LearningPython Nov 7 '12 at 4:50
    
@LearningPython The question is a bit strange. It seems too simple. Maybe you are supposed to see this as a subspace for some other reason? The kernel of some map, the image of another? Any span is a subspace so my answer is true, but maybe there is a more interesting one... –  James S. Cook Nov 7 '12 at 6:08
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