Hello can someone tell me, please, if I anwered the questions correctly.
1- false: The set of rational numbers with the usual addition and multiplication operations by a scalar is a vector space on the real ones.
2-True: All polynomials of degree greater than or equal to 3 with the usual operations are a vector space.
1)
W = {(x, x+1)} is not a subspace of R^2. One reason is that (2,3)+(2,3) = (4,6) is not in W, so it is not closed under addition. Another reason could be that (0,0) is not in W.
2)
W = {p(x) : p(x) in P_n and p(3) = 0}.
These are all single variable polynomials of degree at most n (over some field) that have a factor of (x-3). Note that if you add or scale such polynomials it will still have a factor of (x-3). Note also that the 0 polynomial satisfies 0(3) = 0. So yes, it is a subspace of P_n.
3)
W = {p(x) : p(x) in P_4 and p(2) = 3}
The constant polynomial 3 is in W and 3(2) = 3 by virtue of it being constant.
On the other hand 3+3 = 6 does not satisfy 6(2) = 3, so W is not closed under addition.
Thus W is not a vector subspace of P_4. An easier argument is simply to note that the origin of P_4, i.e. the 0 polynomial, is not an element of W.
Do the rest yourself. A subspace of a vector space V is a subset that is closed under addition, scalar multiplication, and contains the original origin.