Mathematics 44, section 1 -- Linear Algebra

Discussion 2 -- Bases in Vector Spaces

February 5, 1999

Background

Over the last week we have introduced the notions of linear dependence and linear independence for collections of vectors in vector spaces. We have also seen that linearly independent sets are the most "economical" ones to use for making linear combinations and spanning subspaces. This is true since E being linearly dependent is equivalent to L(E) = L(E - {A}) for some A in E so A is, in effect, redundant or unnecessary to make the vectors in L(E). This is the motivation for considering the following new idea:

Definition. Let V be a vector space. A subset E of V is called a basis for V if

1. E spans V -- in other words L(E) = V, and
2. E is linearly independent, so in any equation c₁A₁ + ... + c_nA_n = 0 with c_i R and A_i in E, then all the c_i = 0.

For example, in V = R³, the set E = {(1,0,0), (0,1,0), (0,0,1)} (called the standard basis) is a basis. First, every vector (c₁,c₂,c₃) in R³ can be written as

This shows that R³ = L(E). Second E is clearly linearly independent since if

then c₁ = c₂ = c₃ = 0. Today we want to begin studying bases for vector spaces.

Discussion Questions

• A) Decide whether each of the following sets is a basis of the given vector space:
  1. E = {(1,1,2),(1,-1,0),(3,0,0)} in V = R³.
  2. E = {x - 1, x³ + 3, x + 1, x³ - x} in V = P₃(R).
  3. V = Fun({1,2,3}) = {f : {1,2,3} to R} and E = {f₁,f₂,f₃} where
     f₁(1) = 1, f₁(2) = 0, f₁(3) = 0
     f₂(1) = 0, f₂(2) = 1, f₂(3) = 0
     f₃(1) = 0, f₃(2) = 0, f₃(3) = 1
     
• B) If you have a set E that spans V, can you always find a basis E' for V contained in E? If you can, say how you would find E'; if not, find an example of a spanning set E for some V which does not contain a basis for V.
• C) This problem shows that bases are ``economical'' in a second way as well. Show that if E is a basis for V then for each A in V, there is a { unique way to write A as a linear combination of the vectors in E.
• D) Some vector spaces have no finite basis. For example, this is true of V = Fun(R). Here is an outline for a proof of this fact. Fill in the details by supplying proofs for these statements:
  1. If a vector space V contains an infinite linearly independent subset F, then no basis E of V can be finite. (Arguing by contradiction, suppose you have a finite basis E = {A₁,ldots,A_n} and an infinite linearly independent subset F. You can express each B in F as a linear combination of the A_i. Where does the contradiction come from?)
  2. The following collection of functions in Fun(R) is linearly independent. Let a in R be any real number and define a function f_a by f_a(x) = 1 if x = a, and f_a(x) = 0 if x <> a. Then consider F = { f_a in Fun(R): a in R}.

Assignment

Group write-ups due Wednesday, February 10.