60 CHAPTER 3. VECTOR SPACES

n vectors. If w1 is in the span of the other vectors, delete it. Then consider w2. If it is inthe span of the other vectors, delete it. Continue this way till a shorter list is obtained withthe property that no vector is a linear combination of the others, but its span is still V . ByProposition 3.1.3, the resulting list of vectors is linearly independent and is therefore, abasis since it spans V .

Now suppose for k < n, {u1, · · · ,uk} is linearly independent. Follow the process ofProposition 3.1.8, adding in vectors not in the span and obtaining successively larger lin-early independent sets till the process ends. The resulting list must be a basis. ■

3.3 Exercises1. Show that the following are subspaces of the set of all functions defined on [a,b] .

(a) polynomials of degree ≤ n

(b) polynomials

(c) continuous functions

(d) differentiable functions

2. Show that every subspace of a finite dimensional vector space V is the span of somevectors. It was done above but go over it in your own words.

3. In R2 define a funny addition by (x,y)+ (x̂, ŷ) ≡ (3x+3x̂,y+ ŷ) and let scalar mul-tiplication be the usual thing. Would this be a vector space with these operations?

4. Determine which of the following are subspaces ofRm for some m. a,b are just givennumbers in what follows.

(a){(x,y) ∈ R2 : ax+by = 0

}(b)

{(x,y) ∈ R2 : ax+by≥ y

}(c)

{(x,y) ∈ R2 : ax+by = 1

}(d)

{(x,y) ∈ R2 : xy = 0

}(e)

{(x,y) ∈ R2 : y≥ 0

}(f){(x,y) ∈ R2 : x > 0 or y > 0

}(g) For those who recall the cross product,

{x ∈ R3 : a×x= 0

}.

(h) For those who recall the dot product, {x ∈ Rm : x ·a= 0}(i) {x ∈ Rn : x ·a≥ 0}(j) {x ∈ Rm : x ·s= 0 for all s ∈ S,S ̸= /0,S⊆ Rm} . This is known as S⊥.

5. Show that{(x,y,z) ∈ R3 : x+ y− z = 0

}is a subspace and find a basis for it.

6. In the subspace of polynomials on [0,1] , show that the vectors{

1,x,x2,x3}

are lin-early independent. Show these vectors are a basis for the vector space of polynomialsof degree no more than 3.