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Let $n\ge0$. $P^n$ is a vector space over $\Bbb R$ s.t. it is the set of all polynomials with degree $\le n$ and coefficients in $\Bbb R$.

Consider $W_t=\{f \in P^n : f(1)=t\}$ where $t \in \Bbb R$.

For what values of $t$ is $W_t$ a subspace of $P^n$?

Without giving me the answer, how do I read this question? I'm not sure how to show for which elements $t$ that $W_t$ fulfills the axioms for a subspace because I can't properly interpret what it means by $f(1)=t$. For example: would showing that $f_a(1)+f_b(1)=2t$ (where $f_a(1)=a_1+a_2+...+a_n$ and $f_b(1)=b_1+b_2+...+b_n$) prove closure under addition?

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A subspace must always contain the zero vector. What must $t$ be in order for the zero vector of $P^n$ to lie in $W_t$? –  Henry T. Horton Jan 28 '13 at 0:37
    
But what exactly is the zero vector of $P^n$? –  Duncan Forster Jan 28 '13 at 0:40
    
Can you think of a polynomial $p_0$ such that $p_0+p=p$ for all other polynomials in $P^n$? –  Daniel Rust Jan 28 '13 at 0:41
    
Work with the definition: it's the polynomial $0$ such that $P + 0 = P$ for all polynomials $P$. Can you see what it is? –  Javier Badia Jan 28 '13 at 0:41
    
A polynomial $P_a=a_0+a_1x+a_2x^2+...+a_nx^n$ s.t. $a_0,a_1,...,a_n=0$? –  Duncan Forster Jan 28 '13 at 0:46

2 Answers 2

up vote 2 down vote accepted

You need to check the following:

  1. $0\in W_t$
  2. $\lambda\in\Bbb R,\,f,g\in W_t \implies f+\lambda g\in W_t$.

Now 1. reduces the possible $t$'s, and check 2. for that one.

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The inverse image of a vector subspace under a linear map is linear. Can you use this principle?

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Unfortunately we haven't done linear maps yet, we can only solve this using the axioms. –  Duncan Forster Jan 28 '13 at 0:41

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