Page 61 - Advanced Linear Algebra

Vector Spaces 45

When : ~ ¸# ÁÃÁ# ¹ is a finite set, we use the notation º# Á ÃÁ# » or

span²# ÁÃÁ# ³. A set : of vectors in is said to span = , or generate = , if
=

=~ span ²:³.
It is clear that any superset of a spanning set is also a spanning set. Note also
that all vector spaces have spanning sets, since spans itself.
=
Linear Independence
Linear independence is a fundamental concept.
Definition Let = be a vector space. A nonempty set of vectors in = : is
in ,
:
linearly independent if for any distinct vectors Á Ã Á
bÄb ~ ¬ ~ for all

In words, is linearly independent if the only linear combination of vectors
:
from that is equal to is the trivial linear combination, all of whose
:

coefficients are . If is not linearly independent, it is said to be linearly
:

dependent.
It is immediate that a linearly independent set of vectors cannot contain the zero
vector, since then h ~ violates the condition of linear independence.
Another way to phrase the definition of linear independence is to say that is
:
linearly independent if the zero vector has an “as unique as possible” expression
as a linear combination of vectors from . We can never prevent the zero vector
:
, but we can prevent from

from being written in the form ~ bÄb
being written in any other way as a linear combination of the vectors in .
:
For the introspective reader, the expression ~ b ² c ³ has two

interpretations. One is ~ b where ~ and ~ c , but this does
not involve distinct vectors so is not relevant to the question of linear

independence. The other interpretation is ~ b ! where ! ~c £

(assuming that £ ). Thus, if is linearly independent, then cannot
:
:

contain both and c .

Definition Let be a nonempty set of vectors in . To say that a nonzero
:
=
:
vector #= is an essentially unique linear combination of the vectors in is
to say that, up to order of terms, there is one and only one way to express as a
#
linear combination

# ~ bÄb

where the 's are distinct vectors in and the coefficients are nonzero. More

:

explicitly, #£ is an essentially unique linear combination of the vectors in :
if #º:» and if whenever