Is the set of all polynomials of the form $a_0+a_1x$, where $a_0$ and $a_1$ are real numbers a subspace of $P_3$?
My book says it is not.
Both closure under addition and scalar multiplication hold, so I don't understand why.
And if $P_3$ is taken to mean the set of all polynomials with degree $3$ or less then I don't understand where the problem is.
Best Answer
You are correct, the set of all polynomials of the form $a_0+a_1x$ is indeed a subspace of $\mathcal{P}_3(\mathbf R)$ since it fulfills all the necessary criteria for being a subspace,