[Math] Set of polynomials a subspace of P3

linear algebrapolynomialsvector-spaces

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,

  1. It contains the $0$ vector.
  2. It is closed under addition, $(a_0+a_1x)+(b_0+b_1x)=(a_0+a_1)+(b_0+b_1)x$ which is in that set.
  3. It is closed under scalar multiplication, $\lambda(a_0+a_1x)=(\lambda a_0)+(\lambda a_1)x$ which is in that set.
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