Question

Determine if this is a subspace of the polynomial space
\[p(t) deg \leq n\]

\[p(t)=a+t^2\]

Answer 1

First of all, the zero-vector is not included in this subspace, hence
\[p(t)=0+t^2 \neq 0\]
But if I would continue my test of criteriums (addition and multiplication with scalar), I'm not quite sure what to do.
Addition:
\[\text{Say: } p(t)=a+t^2 \text{ and } q(t)=b+t^2 \text{, then p(t)+q(t)=(p+q)t)}\]
\[p(t)+q(t)=a+t^2+b+t^2=a+b+2t^2\]
\[(p+q)(t)=(a+b)+t^2\neq a+b+2t^2=p(t)+q(t) \text{ <---Does not hold}\]

Multiplication with scalar:
\[a*p(t)=p(a*t)\]
\[a(a+t^2)=a^2+at^2 \neq a+at^2 \text{ <----Does not hold}\]

Can anyone see if I've done this correctly?

Answer 2

If the zero vector is not in the space, then the given space is not a subspace. Every subspace of a vector space contains the vector space's zero vector.

Answer 3

looks good to me