Something a Little Different

Algebra Level 3

Define $P_5$ as the space of all polynomials with degree $5$ , ie of the form $a_1x^5+a_2x^4 \cdots +a_5x+a_6$ where $a_1 \neq 0$ . Is this space a vector space?

No Yes

1 solution

Hamza A
Jun 24, 2019

Solution 1:

Take $p_1=-t^5 \ , \ p_2=t^5$ , both of which are in $P_5$ . $p_1+p_2=0 \notin P_5$ Therefore, $P_5$ is not closed under addition and is not a vector space.

Solution 2:

$P_5$ does not have a zero vector $p_0$ such that $p+p_0 \equiv p$ since $p_0=0\notin P_5$ . Since $P_5$ does not have a zero vector, it is not a vector space.

Something a Little Different

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