Linear Algebra

Section I. Definition of Vector Space 91
1.39 Is this a vector space under the natural operations: the real-valued functions
of one real variable that are differentiable?
1.40 A vector space over the complex numbers C has the same definition as a vector
space over the reals except that scalars are drawn from C insteadoffromR. Show
that each of these is a vector space over the complex numbers. (Recall how complex
numbers add and multiply: (a0 + a1i) +(b0 + b1i) =(a0 + b0) +(a1 + b1)i and
(a0 + a1i)(b0 + b1i) =(a0b0 − a1b1)+(a0b1 + a1b0)i.)
(a) The set of degree two polynomials with complex coefficients
(b) This set
� �
0 a ��
{ a, b ∈ C and a + b =0+0i}
b 0
1.41 Find a property shared by all of the R n ’s not listed as a requirement for a
vector space.
� 1.42 (a) Prove that a sum of four vectors �v1,... ,�v4 ∈ V can be associated in
any way without changing the result.
((�v1 + �v2)+�v3)+�v4 =(�v1 +(�v2 + �v3)) + �v4
=(�v1 + �v2)+(�v3 + �v4)
= �v1 +((�v2 + �v3)+�v4)
= �v1 +(�v2 +(�v3 + �v4))
This allows us to simply write ‘�v1 + �v2 + �v3 + �v4’ without ambiguity.
(b) Prove that any two ways of associating a sum of any number of vectors give
the same sum. (Hint. Use induction on the number of vectors.)
1.43 For any vector space, a subset that is itself a vector space under the inherited
operations (e.g., a plane through the origin inside of R 3 )isasubspace.
(a) Show that {a0 + a1x + a2x 2 � � a0 + a1 + a2 =0} is a subspace of the vector
space of degree two polynomials.
(b) Show that this is a subspace of the 2×2 matrices.
� �
a b ��
{ a + b =0}
c 0
(c) Show that a nonempty subset S of a real vector space is a subspace if and only
if it is closed under linear combinations of pairs of vectors: whenever c1,c2 ∈ R
and �s1,�s2 ∈ S then the combination c1�v1 + c2�v2 is in S.
2.I.2 Subspaces and Spanning Sets
One of the examples that led us to introduce the idea of a vector space
was the solution set of a homogeneous system. For instance, we’ve seen in
Example 1.4 such a space that is a planar subset of R 3 . There, the vector space
R 3 contains inside it another vector space, the plane.
2.1 Definition For any vector space, a subspace is a subset that is itself a
vector space, under the inherited operations.