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Is the set $W = \{f(x) \in P(F) : f(x) = 0 \text{ or } f(x) \text{ has degree } n\}$ a subspace of $P(F)$ if $n\geq 1$? In order for this subset to be a subspace we have to make sure it satisfies the three conditions:

  1. $0\in W$

  2. $x+y\in W$ whenever $x\in W$ and $y\in W$

  3. $cx\in W$ whenever $c\in F$ and $x\in W$

I believe that the first condition is already fulfilled because we're given a condition that $f(x)=0$.

Next we have to see if set is closed under addition and scalar multiplication.

Can you please help me from here?

I believe it is not closed under addition but I don't know how to show how.

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What about the sum of $X$ and $1-X$? – Hagen von Eitzen Jan 16 '13 at 7:12
What does $F$ stand for? What does $P(F)$ mean? Please edit the body of the question so that these notations are explained therein. – Gerry Myerson Jan 16 '13 at 11:13

If $P(F)$ means the space of polynomials over a field $F,$ then it should be rewritten as $F[x].$ As you said, the set $W = \{f(x) \in F[x] \mid f = 0 \textrm{ or } \deg f = n\}$ is not closed under addition when $n \ge 1.$ Let $a x^n + b x^{n-1} + c \in W,$ where $a,b \not = 0.$ Then the polynomial $-a x^n - c \in W$ as well, but their sum is $bx^{n-1} \notin W,$ so $W$ is not a subspace of $F[x].$

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