(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For each of the following subsets U of the vector space V decide whether or not U is a

subspace of V . Give reasons for your answers. In each case when U is a subspace, find a

basis for U and state dim U

2. Relevant equations

[tex]V=P_{3} ; U=\left\{p\in\ P_{3}:p(0)-p'(0)=0\right\}[/tex]

3. The attempt at a solution

so i set it out as (a_3p^3+...+a_1p^1+a_0)-(3a_3p^2+2a_2p+a_1)=0 then rearranged all that to get; a_3p^3+...+(a_1-2a_2)p^1+(a_0-a_1)=0

Now i'm not sure what to do next. I think its because i'm not comfortable with the idea of polynomials as regards to vector spaces. does the fact that a couple of them share coefficients as it were mean that they would be dependent and so reduce the dimension? i guess making it a one dimensional subspace? All help and explanations much appreciated.

p.s. another subset is [tex]x=(x_{1},x_{2},x_{3}): x^{2}_{3}=x^{2}_{1}+x^{2}_{2}[/tex]

I want to say this is not a subspace because of the fact its not a linear equation? would that be correct? Doesn't feel like very rigorous reasoning so it feels wrong.

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# Homework Help: Vector spaces and subspaces

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