Linear Algebra

Section I. Definition of Vector Space 95
can take x − 2y + z = 0 to be a one-equation linear system and expressing the
leading variable in terms of the free variables x =2y− z.
⎛ ⎞
2y − z
S = { ⎝ y ⎠
z
� ⎛
� y, z ∈ R} = {y ⎝ 2
⎞ ⎛
1⎠
+ z ⎝
0
−1
⎞
0 ⎠
1
� � y, z ∈ R}
Now the subspace is described as the collection of unrestricted linear combinations
of those two vectors. Of course, in either description, this is a plane
through the origin.
2.12 Example This is a subspace of the 2×2 matrices
� �
a 0 ��
L = { a + b + c =0}
b c
(checking that it is nonempty and closed under linear combinations is easy). To
paramatrize, express the condition as a = −b − c.
� � � � � �
−b − c 0 �� −1 0 −1 0 ��
L = {
b, c ∈ R} = {b + c
b, c ∈ R}
b c
1 0 0 1
As above, we’ve described the subspace as a collection of unrestricted linear
combinations (by coincidence, also of two elements).
Paramatrization is an easy technique, but it is important. We shall use it
often.
2.13 Definition The span (or linear closure) of a nonempty subset S of a
vector space is the set of all linear combinations of vectors from S.
�
[S] ={c1�s1 + ···+ cn�sn � c1,... ,cn ∈ R and �s1,... ,�sn ∈ S}
The span of the empty subset of a vector space is the trivial subspace.
No notation for the span is completely standard. The square brackets used here
are common, but so are ‘span(S)’ and ‘sp(S)’.
2.14 Remark In Chapter One, after we showed that the solution set of a
homogeneous linear system can written as {c1 � β1 + ···+ ck � �
βk
� c1,... ,ck ∈ R},
we described that as the set ‘generated’ by the � β’s. We now have the technical
term; we call that the ‘span’ of the set { � β1,... , � βk}.
Recall also the discussion of the “tricky point” in that proof. The span of
the empty set is defined to be the set {�0} because we follow the convention that
a linear combination of no vectors sums to �0. Besides, defining the empty set’s
span to be the trivial subspace is a convienence in that it keeps results like the
next one from having annoying exceptional cases.
2.15 Lemma In a vector space, the span of any subset is a subspace.