Linear Algebra, 2020a

Section I. Definition of Vector Space 101
is a linear combination of elements of S and so is an element of [S] (possibly
some of the ⃗s i ’s from ⃗v equal some of the ⃗s j ’s from ⃗w but that does not matter).
QED
The converse of the lemma holds: any subspace is the span of some set,
because a subspace is obviously the span of itself, the set of all of its members.
Thus a subset of a vector space is a subspace if and only if it is a span. This
fits the intuition that a good way to think of a vector space is as a collection in
which linear combinations are sensible.
Taken together, Lemma 2.9 and Lemma 2.15 show that the span of a subset
S of a vector space is the smallest subspace containing all of the members of S.
2.16 Example In any vector space V, for any vector ⃗v ∈ V, the set {r · ⃗v | r ∈ R}
is a subspace of V. For instance, for any vector ⃗v ∈ R 3 the line through the
origin containing that vector {k⃗v | k ∈ R} is a subspace of R 3 . This is true even
if ⃗v is the zero vector, in which case it is the degenerate line, the trivial subspace.
2.17 Example The span of this set is all of R 2 .
( ) ( )
1 1
{ , }
1 −1
We know that the span is some subspace of R 2 . To check that it is all of R 2 we
must show that any member of R 2 is a linear combination of these two vectors.
So we ask: for which vectors with real components x and y are there scalars c 1
and c 2 such that this holds?
( ) ( ) ( )
1 1 x
c 1 + c 2 =
(∗)
1 −1 y
Gauss’s Method
c 1 + c 2 = x
c 1 − c 2 = y
−ρ 1 +ρ 2
−→
c 1 + c 2 = x
−2c 2 =−x + y
with back substitution gives c 2 =(x − y)/2 and c 1 =(x + y)/2. This shows
that for any x, y there are appropriate coefficients c 1 ,c 2 making (∗) true — we
can write any element of R 2 as a linear combination of the two given ones. For
instance, for x = 1 and y = 2 the coefficients c 2 =−1/2 and c 1 = 3/2 will do.
Since spans are subspaces, and we know that a good way to understand a
subspace is to parametrize its description, we can try to understand a set’s span
in that way.
2.18 Example Consider, in the vector space of quadratic polynomials P 2 , the
span of the set S = {3x − x 2 ,2x}. By the definition of span, it is the set of