4.2

Question 1

4. 2.1 : If W is a set of one or more vectors in a vector space V, then W is a subspace of V if and only if the following conditions hold
   a) If u and v are vectors in W, then u + v is in W
   b) If k is any scalar and u is any vector in W, then ku is in W

Answer 1

If W is a subspace of V, then all the vector space axioms hold in W, including 1 and 6.
Let u be any vector in W. it follows that both 0u = 0 and (-1)u = -u are in W.

Question 2

4. 2.3: If S = (w1, w2, …, wr) is a nonempty set of vectors in a vector space V, then:
   a) The set W of all possible linear combinations of the vectors in S is a subspace of V
   b) The set W in part a is the smallest subspace of V that contains all of the vectors in S in the sense that any other subspace that contains those vectors contains W

Answer 2

proof a: Let W be the set of all possible linear combinations of the vectors in S. We must show that S is closed under addition and scalar multiplication. To prove closure under addition, let
u = c1w1 + c2w2 + … + crwr and v = k1w1 + k2w2 + … + krwr
be 2 vectors in S.it follows that their sum can be written as … which is a linear combination of the vectors in S.

Question 3

4.2.4: The solution set of a homogeneous linear system Ax = 0 in n unknowns is a subspace of Rn.

Answer 3

prove that it is closed under addition and scalar multiplication. p.187