Mathematics 44, section 1 -- Linear Algebra

Discussion 1 -- Vector Subspaces, Spans, etc.

January 27, 1999

Background

In class on Monday, we defined a vector subspace of a vector space V to be a non-empty subset W of V satisfying:

1. If A,B are elements of W, then A + B is also an element of W (where + is the vector sum operation in V)
2. If A is in W and t is in R, then t.A is an element of W (where . is the scalar multiplication operation from V).

We saw that a vector subspace W of V is itself a vector space under +,. from V.

Discussion Questions

• A) Let V = R⁴ and W = {(x₁,x₂,x₃,x₄) in R⁴ : 3x₁ - 2x₂ + x₄ = 0 and x₁ + 3x₂ - x₃ = 0}.
  1. Show that W is a vector subspace of V.
  2. How is W related to the two subspaces: W₁ = {(x₁,x₂,x₃,x₄) in R⁴ : 3x₁ - 2x₂ + x₄ = 0} and W₁ = {(x₁,x₂,x₃,x₄) in R⁴ : x₁ +3x₂ - x₃ = 0}?
  3. Show in general that if W₁ and W₂ are vector subspaces of V, then W₁ intersect W₂ is also a vector subspace of V.
• B) Let V be a vector space, and let E in V be a subset. At the end of class on Monday, we introduced the idea of a linear combination of elements of E -- a linear combination of elements of E is any finite sum of the form
  
  c₁ A₁ + c₂ A₂ + ... + c_n A_n,

  where A_i is in E and c_i is in R for all i = 1,..., n. We also introduced the notation:
  

  L(E) = {c₁ A₁ + c₂ A₂ + ... + c_n A_n : A_i is in E, c_i is in R }

  for the collection of all linear combinations of elements of E. We call L(E) the linear span of E.

  1. Show that if V = Fun(S) for S = (-infty,+infty) = R, and W = { f in Fun(S) : f'(x),f''(x) exist for all} x, f''(x) - 3f'(x) + 2f(x) = 0} then W is a vector subspace of V.
  2. Show that L({e^x, e^2x}) is contained in W.
  3. Show that L({e^x, e^2x}) is also a vector subspace of V.
  4. Generalizing what you did in part 3, show that L(E) is a vector subspace of V whenever E is a non-empty subset of V. If we also wanted the linear span of the empty set to be a vector subspace of V, how should we define the linear span of the empty set?
  5. Is sin² x in L({1,sin x, cos x, sin 2x, cos 2x})? What about e^x? Why or why not? (Be careful!)

Assignment

Group write-ups due in class, Monday February 1.