Which of the following are subspaces of P2 with usual operat

Which of the following are subspaces of P_2 with usual operations? {1 +ax: x belongs to R} span {1 +x, 1+ x^2} span {1, x, x^2, x^3} = P_3

Solution

1.

Consider two elements:

1+ax,1+ay with x,y in R

For this set to be subspace sum of theset two elements must be in the set

1+ax+1+ay=2+a(x+y) which is not in the set

Hence not a subspace

2.

Let, P,Q be in this set so : a,b,r,s are scalars so that

P=a(1+x)+b(1+x^2)

Q=r(1+x)+s(1+x^2)

P+Q=(a+r)(1+x)+(b+s)(1+x^2)

which is in the set

Let, P be in the set as before and c be a real number

cP=ca(1+r)+cb(1+x^2)

Hence, cP is in the set

Hence it is a subspace

3. No it is not a subpace as P2 is properly contained in this set