Consider the set P3 of polynomials in x epsilon R of degree

Consider the set, P_3, of polynomials in x epsilon R of degree 3. So, P_3 = {a_0 + a_1x + a_2x^22 + a_3x^3 for a_0 epsilon R, a_1 epsilon R, a_2 epsilon R, a_3 epsilon R}. Is P_3 a vector space? Why or why not? If so, what is it\'s dimension? If so, what is a basis?

Solution

a)Yes it is a vector space as it satisfies following three properties

1. 0 belongs to this set

2. Adding any two polynomials of degree 3 or less gives a polynomial of degree 3 or less hence again in the set so it is closed undre addition

3. Multiplying any polynomial does not change its degree hence closed under scalar multiplication

Hence a vector space

b)

Standard basis is

1,x,x^2,x^3 which are clearly linearly independent and 4 in number hence dimension is 4

We can prove linear independence by simply setting

a+bx+cx^2+dx^3=0

HEnce, a=b=c=d=0

c)

Basis is {1,x,x^2,x^3}