A First Course in Linear Algebra, 2017a

9.4. Subspaces and Basis 477
(b) { x 3 + 1,x 2 + x,2x 3 + x 2 ,2x 3 − x 2 − 3x + 1 }
Exercise 9.3.32 In the context of the above problem, consider polynomials
{
ai x 3 + b i x 2 + c i x + d i , i = 1,2,3,4 }
Show that this collection of polynomials is linearly independent on an interval [s,t] if and only if
⎡
⎤
a 1 b 1 c 1 d 1
⎢ a 2 b 2 c 2 d 2
⎥
⎣ a 3 b 3 c 3 d 3
⎦
a 4 b 4 c 4 d 4
is an invertible matrix.
Exercise 9.3.33 Let the field of scalars be Q, the rational numbers and let the vectors be of the form
a + b √ 2 where a,b are rational numbers. Show that this collection of vectors is a vector space with field
of scalars Q and give a basis for this vector space.
Exercise 9.3.34 Suppose V is a finite dimensional vector space. Based on the exchange theorem above, it
was shown that any two bases have the same number of vectors in them. Give a different proof of this fact
using the earlier material in the book. Hint: Suppose {⃗x 1 ,···,x ⃗ n } and {⃗y 1 ,···,y ⃗ m } are two bases with
m < n. Then define
φ : R n ↦→ V ,ψ : R m ↦→ V
by
φ (⃗a)=
n ( ) m
∑ a k ⃗x k , ψ ⃗b = ∑ b j ⃗y j
k=1
j=1
Consider the linear transformation, ψ −1 ◦ φ. Argue it is a one to one and onto mapping from R n to R m .
Now consider a matrix of this linear transformation and its reduced row-echelon form.
9.4 Subspaces and Basis
Outcomes
A. Utilize the subspace test to determine if a set is a subspace of a given vector space.
B. Extend a linearly independent set and shrink a spanning set to a basis of a given vector space.
In this section we will examine the concept of subspaces introduced earlier in terms of R n . Here, we
will discuss these concepts in terms of abstract vector spaces.
Consider the definition of a subspace.