Which of these sets are subspaces of $P_3$

linear algebramatrices

Which of the following sets is a subspace of $P_{3}$?

a. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = 0 \text{ and } a_2 = a_3 \}$

b. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = 1 \text{ and } a_2 = 2a_3 \}$

c. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = a_2^2 \}$

d. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_0 + 2a_1 – 3a_2 = 2 \}$

I am confused about this question because aren't all of these subspaces of $P_3$? All of these polynomials have a degree of less than or equal to 3, so shouldn't they all be entirely contained in the set $P_3$?

Best Answer

They are all subsets of $P_3$, if that's your notation for the polynomials of degree $\le 3$, but they are not all linear subspaces. For a subset $S$ of a vector space to be a linear subspace, it needs two things to be true:

  1. For every $u$ and $v$ in $S$, $u + v$ is also in $S$.
  2. For every $u$ in $S$ and every scalar $a$, $a u$ is also in $S$.
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