09 Subspaces, Spans, and Linear Independence
Chapter Two, Sections 1.II and 2.I look at several different
kinds of subset of a vector space.
A subspace of a vector space
is a subset of that is itself a vector space, using the vector
addition and scalar multiplication that are inherited from .
(This means that for and in the subspace and a real
number , and have the same
values whether and are thought of as elements
of the subspace as they do when and are thought of
as elements of )
Suppose is a vector space and , and you want
to show that is a subspace of . This means verifying that
satisfies the ten properties listed in the definition
of vector space. However, satisfies many of those properties
automatically, because they are already known to hold in .
For example, if , we know without checking it
that because that equation is true
for all elements of . To show that is a subspace of ,
it suffices to check just three things:
- is closed under vector addition. That is for all ,
.
- is closed under scalar multiplication. That is, for all
and , .
- is not empty. (Usually, this is shown by showing .)
It's usually fairly easy to check that a subset is a subspace.
For example the set of differentiable functions from to
is a subspace of the vector space, , of all functions from
to . This is because (1) the sum of two differentiable functions
is differentiable; (2) a constant times a differentiable function is
differentiable; and (3) the constant function (which is the
additive identity in ) is differentiable.
If you take some arbitrary subset of a vectors space , it is
probably not a subspace. However, you can "generate" a subspace
from any subset of by taking the "span" of that subset.
Definition:
Suppose that is a vector space, and is any non-empty
subset of . The span of is defined to
be the subset, , of given by
That is, is the set of all linear combinations of any number of vectors in .
The notation is also used instead of .
In addition, wel say that ; that is, the span of the
empty set is the trivial subspace that contains only the single vector
Theorem:
If is a vector space and , then the span, , of
is a subspace of .
Proof:
If , then , which is a subspace of . Suppose that is non-empty.
Then
- is not empty. For, since , there is some . Now, is a linear combination
of the single vector , so by defintion of . So .
[Since , we see that . This shows that .]
- is closed under vector addition. For suppose that . We must show
that . But, by definition of , we can write
and ,
were and .
But then, ,
which is in since it is a linear combination of elements of . [Note that we do not require the
vectors in a linear combination to be distinct, so it's OK if some of the are equal to
some of the .]
- is closed under scalar multiplication. For let and .
Write where and
. Then , which is in since it is a linear combination of
elements of .
Since has these three properties, it is a subspace.
If , we say that spans or
generates , and that is a spanning set
for
We have actually been working with spans for a while. If consists of a single non-zero
vector , then is the set of all scalar multiples of . That is,
. So the span of is the line through the origin in direction
. If we add another vector, , to , the span of would be
. If is a multiple of , then it adds nothing new to the span;
you still just get the line through the origin in direction . But if does not lie
on that line, then and generate a plane that contains the origin.
If we then add a third vector, , to , the possibilities are that either lies
on the plane spanned by and (in which case, it adds nothing new to the span),
or is not on the plane (in which case , , and span a three-dimensional
linear space). I will also note that the solution set of a homogeneous linear system written
as a set is the
span of the set . In particular, it is a subspace.
It is also easy to prove that fact directly.
Note by the way that when we take the span of a finite set of vectors, say
, then the span can be written
.
That is, we can include all vectors from in every linear combination in the span,
even though the definition allows linear combinations with any number of vectors from .
This is because some or all of the coefficients, , can be zero in a linear combination,
and because if the same vector is used several times in a linear combination, all the
terms that use it can be added together to get a single term (for example,
can be combined to make ).
The idea that adding another vector to a spanning set will add something new to the
span, unless that new vector is a linear combination of the vectors already in the set,
is an essential one. It leads to the idea of linear dependence and linear independence.
Spanning and linear independence are closely related concepts that are absolutely
central to linear algebra.
Definition: Let be a vector
space, and let be a subset of . We say that is linearly
independent (or that the vectors in are linearly independent) if
no element of can be written as a linear combination of other elements of .
A set that is not linearly independent is called linearly dependent.
I actually prefer a different definition of linear dependence and independence,
and I believe that this alternative definition is more common in math books.
Definition: Let be a vector
space, and let be a subset of . We say that is linearly
independent if in any linear combination that adds up to zero,
where , all of the coefficients must be zero.
And is linearly dependent if there is a linear combination of elements of that
adds up to , and in which at least one of the coefficients is non-zero.
I will accept either definition. It is easy to prove that the two
definitions are equivalent.
But there is another complication. Sometimes, instead of dealing with sets
of vectors, we work with lists of vectors. The difference is that a list
is ordered and can contain duplicates while a set cannot contain duplicates and
the order of the elements in a set does not matter. So, for example,
and are different lists, but and are
the same set. And and are different lists while
is just another way of writing the set . Order is not really important
for linear independence, but there is a subtle difference between linear independence
of lists and linear independence of sets that arises because of the possibility
of duplicates in a list. Consider this definition, and note this in this case,
all of the elements from the list are used in the linear combination:
Definition: Let be a vector
space, and let be a list of elements of .
We say that this list of vectors is linearly
independent if in any linear combination that adds up to zero,
all of the coefficients must be zero.
And the list is linearly dependent if there is a linear combination of this form that
adds up to , and in which at least one of the coefficients is non-zero.
The subtle difference turns up when you consider a list that contains
duplicates. A list of vectors that contains duplicates cannot be linearly independent.
For example, the list is linearly dependent because
. However, the set
is really just the set , which is linearly independent unless one of
the two vectors is a multiple of the other. Usually, it is clear whether we are working
with lists or with sets of vectors.
Many things can be proved about linearly independent sets and their spans. Mostly
they amount to understanding that a set is linearly independent if it is a
minimal spanning set for the subspace that it spans. That is, removing any element
from a linearly independent set will remove some things from the span. On the
other hand, if a set is linearly dependent, then there is at least one element
that you can remove without affecting the span. That would be an element that
is a linear combination of other elements from the set. That element adds
nothing new to the span of the other elements.
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