Definition. A subspace of a vector space V is a subset H of V that has three properties
(a) The zero vector of V is in H.
(b) $\forall$ u, v $\in H$, we have u + v $\in H$.
(c) $\forall$ u $\in H$, c scalar, we have c u $\in H$.
For $n \geq$ 0, denote $\mathbb{P}_n$ to be the set of all polynomials of degree at most $n$. Which of the following are subspaces of $\mathbb{P}_n$ for an appropriate value of $n$? Support your answer with reason.
(a) All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$.
(b) All polynomials of degree at most 3, with integers as coefficients.
Solution:
Both the statements (a) and (b) above are not subspaces of $\mathbb{P}_n$.
Reason:
For part (a), All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$, is not a subspace because it failed to meet the standards of the first property stated above, that is, the coefficient of the $t^2$ term is always a 1, therefore the zero vector is not in this subspace of polynomials.
For part (b), All polynomials of degree at most 3, with integers as coefficents is also not a subspace becuase it as well failed to meet the standards of the third property, that is, integer coefficents, when multiplied by a real scalar, will not necessarily be of integer type leading to integer coefficents. Thusly, the resulting polynomial will not be in the subset of polynomials $\mathbb{P}_n$. $\Box$
Can someone please look at this and see if I have the right idea and approach to this question. Thank You
Best Answer
Yes, you are right in both cases.