Tensor product of vector spaces
Let
K
{\displaystyle {}K}
field , and let
V
1
,
…
,
V
n
{\displaystyle {}V_{1},\ldots ,V_{n}}
K
{\displaystyle {}K}
vector spaces .
Let
F
{\displaystyle {}F}
K
{\displaystyle {}K}
vector space
generated by all symbols
(
v
1
,
…
,
v
n
)
{\displaystyle {}(v_{1},\ldots ,v_{n})}
v
i
∈
V
i
{\displaystyle {}v_{i}\in V_{i}}
e
(
v
1
,
…
,
v
n
)
{\displaystyle e_{(v_{1},\ldots ,v_{n})}}
U
⊆
F
{\displaystyle {}U\subseteq F}
K
{\displaystyle {}K}
linear subspace
of
F
{\displaystyle {}F}
r
e
(
v
1
,
…
,
v
i
−
1
,
v
i
,
v
i
+
1
,
…
,
v
n
)
−
e
(
v
1
,
…
,
v
i
−
1
,
r
v
i
,
v
i
+
1
,
…
,
v
n
)
{\displaystyle {}re_{(v_{1},\ldots ,v_{i-1},v_{i},v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},rv_{i},v_{i+1},\ldots ,v_{n})}}
e
(
v
1
,
…
,
v
i
−
1
,
u
+
w
,
v
i
+
1
,
…
,
v
n
)
−
e
(
v
1
,
…
,
v
i
−
1
,
u
,
v
i
+
1
,
…
,
v
n
)
−
e
(
v
1
,
…
,
v
i
−
1
,
w
,
v
i
+
1
,
…
,
v
n
)
{\displaystyle {}e_{(v_{1},\ldots ,v_{i-1},u+w,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},u,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},w,v_{i+1},\ldots ,v_{n})}}
Then the
residue class space
F
/
U
{\displaystyle {}F/U}
tensor product
V
i
{\displaystyle {}V_{i}}
i
∈
{
1
,
…
,
n
}
{\displaystyle {}i\in \{1,\ldots ,n\}}
V
1
⊗
K
V
2
⊗
K
⋯
⊗
K
V
n
.
{\displaystyle V_{1}\otimes _{K}V_{2}\otimes _{K}\cdots \otimes _{K}V_{n}.}
