Answer to Question #294872 in Linear Algebra for Yamuna N

Question

For p∈P3(R) given by p(x)=a0+a1x+⋯+a3x3, let s(p)=a0+a1+a2+a3 and det(p)=a0. Also, corresponding to the polynomial p∈P3(R), we define the polynomial p∗ to be p(−x). Which of the following are subspaces of P3(R) ?

Answer 1

Subspace of $P_3(R)$ satisfies three conditions

a) Contain zero vector

b) closed under addition

c) closed under scalar multiplication

Zero Vector

$\forall_a\in\R$ such that $a=0$

$P(R)=0$ is part of set

For any $P(R):P(R)+0=P(R)$

$\therefore$ The set contain zero vector

Vector addition

For any two polynomials $a_1x^3$ and $a_2x^3$

$a_1x_3+a_2x_3=(a_1+a_2)x^3$

$=kx^3\in\>ax^3$

So it is closed under vector addition

Scalar multiplier

Chosing arbitrary polynomial $ax^3$

$b.ax^3=bax^3\in\>ax^3$

Therefore a subspace of $P_3(R)$ is closed under scalar multiplication