Plan

• 1.3
• 1.4

1.3

• Definition: A subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

• Problem: Normally, there are 8 properties you need to check. But it turns out you only need to check 4 of them. Which 4? Why?

• Problem: (Theorem 1.3) Let V be a vector space and W and subset of V. Then W is a subspace of V if and only if the following three conditions hold for the operations defined in V.
  1. 0 ∈ W.
  2. x + y ∈ W whenever x ∈ W and y ∈ W.
  3. cx ∈ W whenever c ∈ F and x ∈ W.
• Problem: Same problem as before, but replace conditions 2 and 3 with
  • cx + y ∈ W whenever, x, y ∈ W and c ∈ F.

• Problem: Give an example of a vector space V and a subset W of V such that, W is a vector space but W is not a subspace of V.

• Problem: Show that the intersection of 2 subspaces is a subspace.

1.4

• Definition: Let V be a vector space and S a nonempty subset of V. A vector v ∈ V is called a linear combination of vectors of S if there exists a finite number of vectors u₁, …, u_n in S and scalars a₁, …, a_n in F such that v = a₁u₁ + ⋯ + a_nu_n.

• Problem: We denote the set of all linear combinations of S by spanS. By convention, we define the span of the empty set to be the trivial subspace {0}. Prove that span(S) is always a subspace.

• Problem: Let S ⊆ T be sets inside of a vector space V. Prove that span(S) is a subspace of span(T).

• Problem: Prove that span(S) is a the smallest subspace containing S. (This gives an alternative definition of span(S) that turns out to be quite useful!)