The construction of the dual basis is a special case of the construction in the proof of
2.4; take
W
=
R
with the single basis element
w
1
= 1; then the
v
*
i
of (2.6.1.1) is equal to
the
f
i
1
of (2.4.1).
I7
Therefore a choice of basis
{
v
i
}
yields an explicit isomorphism
φ
:
V
→
V
*
, namely
φ
(
v
i
) =
v
*
i
(2
.
6
.
1
.
2)
((2.6.1.2) defines
φ
uniquely by (1.9).)
Note that this isomorphism depends on the choice of basis; if we had chosen a different
basis for
V
, we would have constructed a different isomorphism.
There is no good way to select a “preferred” isomorphism from
V
to
V
*
; we will revisit
this issue in Section 5 of this chapter.
Remark 2.6.1.3
The element
v
*
i
of the dual basis depends not just on
v
i
but on
all the
v
j
. If
{
w
1
, . . . , w
n
}
is another basis and
v
1
=
w
1
, it does
not
follow that
v
*
1
=
w
*
1
.
Definition 2.7.
The
double dual
of
V
is the vector space
V
**
= (
V
*
)
*
.
2.8.
Applying (2.6) twice and invoking (1.4), we can construct an isomorphism
V
→
V
**
. But there is a simpler, basis independent construction. Map
φ
:
V
→
V
**
by
φ
(
v
)(
g
) =
g
(
v
)
and check that this map is oneone and onto.
2B. Multilinearity
Definition 2.9.
Let
V
,
W
and
U
be vector spaces. Then a
bilinear map
f
:
V
×
W
→
U
is a function satisfying the following axioms for
v
i
∈
V
,
w
i
∈
W
and
α
∈
R
:
I8
i)
f
(
αv
1
+
v
2
, w
) =
αf
(
v
1
, w
) +
f
(
v
2
, w
)
ii)
f
(
v, αw
1
+
w
2
) =
f
(
v, w
1
) +
αf
(
v, w
2
)
Remark 2.9.1.
You can think of multilinearity as “linearity in each variable sepa
rately”. More precisely, we can restate conditions (2.9(i)) and (2.9(ii)) as follows:
i
0
) For each fixed
v
∈
V
, the map
f
v
:
W
→
U
defined by
f
v
(
w
) =
f
(
v, w
)
is linear.
ii
0
) For each fixed
w
∈
W
, the map
f
w
:
V
→
U
defined by
f
w
(
v
) =
f
(
v, w
)
is linear.
Remark 2.2.1.2.
As sets,
V
×
W
is the same thing as
V
⊕
W
, so you might be
tempted to think of a bilinear map as a function
f
:
V
⊕
W
→
U
But it is important to realize that this map is
not
in general a linear transformation. Here’s
why: Applying axioms 2.9(i) and 2.9(ii), we find that for a bilinear map
f
we have
f
(
v
1
+
v
2
, w
1
+
w
2
) =
f
(
v
1
, w
1
) +
f
(
v
1
, w
2
) +
f
(
v
2
, w
1
) +
f
(
w
1
, w
2
)
whereas for a linear transformation
f
we have
f
(
v
1
+
v
2
, w
1
+
w
2
) =
f
(
v
1
, w
1
) +
f
(
v
2
, w
2
)
I9
which is not at all the same thing.
We will want to adopt the convention that whenever we write down a map between
vector spaces, it is assumed to be linear. Therefore, when
f
is a bilinear map, we will think
of its domain as the
set
of ordered pairs
V
×
W
, rather than the
vector space
of ordered
pairs
V
⊕
W
.
Definition 2.10.
We generalize the notion of bilinearity as follows: A map
f
:
V
1
×
V
2
× · · · ×
V
n
→
W
is called
multilinear
if it is linear in each variable separately. More precisely, the condition
is that if we fix elements in
n

1 of the vector spaces
V
1
, . . . V
n
, then the induced map on
the remaining
V
i
is linear.
2C. Tensor Products
2.11.
The
tensor product
of two vector spaces
V
and
W
is a vector space
V
⊗
W
which is the recipient of a “universal” multilinear map
t
:
V
×
W
→
V
⊗
W
(2
.
11
.
1)
“Universal” means that any multilinear map with domain
V
×
W
can be factored
uniquely as a composition
g
◦
t
where
t
:
V
⊗
W
→
U
is a
linear
map.
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 Fall '14
 Linear Algebra, Algebra, Addition, Multiplication, Scalar, Vector Space, Functors