1. Let V be the set of all pairs (x; y) of real numbers and suppose vector
addition and scalar multiplication are defined in the following way:
(x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
a(x, y) = (ax, y),
for any scalar a in the field of real numbers. Is the set V , with these operations,
a vector space over the field of real numbers? Justify your answer.
2. Let W1 and W2 be subspaces of a vector space V such that their union W1 U W2 (W1 U W2 is the set of vectors which belong to either W1 or W2) is also
a subspace of V . Prove that either W1 is contained in W2 or vice versa.