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Transcript
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“main”
2007/2/16
page 270
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270
CHAPTER 4
Vector Spaces
Theorem 4.5.6
Let {v1 , v2 , . . . , vk } be a set of at least two vectors in a vector space V . Then {v1 , v2 , . . . , vk }
is linearly dependent if and only if at least one of the vectors in the set can be expressed
as a linear combination of the others.
Proof If {v1 , v2 , . . . , vk } is linearly dependent, then according to Definition 4.5.3, there
exist scalars c1 , c2 , . . . , ck , not all zero, such that
c1 v1 + c2 v2 + · · · + ck vk = 0.
Suppose that ci = 0. Then we can express vi as a linear combination of the other vectors
as follows:
vi = −
1
(c1 v1 + c2 v2 + · · · + ci−1 vi−1 + ci+1 vi+1 + · · · + ck vk ).
ci
Conversely, suppose that one of the vectors, say, vj , can be expressed as a linear combination of the remaining vectors. That is,
vj = c1 v1 + c2 v2 + · · · + cj −1 vj −1 + cj +1 vj +1 + · · · + ck vk .
Adding (−1)vj to both sides of this equation yields
c1 v1 + c2 v2 + · · · + cj −1 vj −1 − vj + cj +1 vj +1 + · · · + ck vk = 0.
Since the coefficient of vj is −1 = 0, the set of vectors {v1 , v2 , . . . , vk } is linearly
dependent.
As far as the minimal-spanning-set idea is concerned, Theorems 4.5.6 and 4.5.2 tell
us that a linearly dependent spanning set for a (nontrivial) vector space V cannot be a
minimal spanning set. On the other hand, we will see in the next section that a linearly
v3
independent spanning set for V must be a minimal spanning set for V . For the remainder
v2
of this section, however, we focus more on the mechanics of determining whether a given
v1
set of vectors is linearly independent or linearly dependent. Sometimes this can be done
x
by inspection. For example, Figure 4.5.2 illustrates that any set of three vectors in R2 is
linearly dependent.
Figure 4.5.2: The set of vectors
As another example, let V be the vector space of all functions defined on an interval
{v1 , v2 , v3 } is linearly dependent
I
.
If
2
in R , since v3 is a linear
f1 (x) = 1,
f2 (x) = 2 sin2 x,
f3 (x) = −5 cos2 x,
combination of v1 and v2 .
y
then {f1 , f2 , f3 } is linearly dependent in V , since the identity sin2 x + cos2 x = 1
implies that for all x ∈ I ,
f1 (x) = 21 f2 (x) − 15 f3 (x).
We can therefore conclude from Theorem 4.5.2 that
span{1, 2 sin2 x, −5 cos2 x} = span{2 sin2 x, −5 cos2 x}.
In relatively simple examples, the following general results can be applied. They are a
direct consequence of the definition of linearly dependent vectors and are left for the
exercises (Problem 49).
Proposition 4.5.7
Let V be a vector space.
1. Any set of two vectors in V is linearly dependent if and only if the vectors are
proportional.
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