definition of tensor product of \(k\) vectors spaces over field
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of product set.
- The reader knows a definition of free vectors space on set.
- The reader knows a definition of sub-'vectors space' generated by subset of vectors space.
- The reader knows a definition of quotient vectors space of vectors space by sub-'vectors space'.
Target Context
- The reader will have a definition of tensor product of \(k\) vectors spaces over field.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_1, ..., V_k\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( V_1 \times ... \times V_k\): \(= \text{ the product set }\)
\( F (V_1 \times ... \times V_k, F)\): \(= \text{ the free vectors space }\)
\( S\): \(= \{((v_1, ..., r v_j, ..., v_k)) - r ((v_1, ..., v_k)) \in F (V_1 \times ... \times V_k) \vert r \in F, v_1 \in V_1, ..., v_k \in V_k\} \cup \{((v_1, ..., v_j + v'_j, ..., v_k)) - ((v_1, ..., v_j, ..., v_k)) - ((v_1, ..., v'_j, ..., v_k)) \in F (V_1 \times ... \times V_k) \vert v_1 \in V_1, ..., v_k \in V_k, v'_j \in V_j\}\)
\( (S)\): \(= \text{ the sub-'vectors space' generated by subset of vectors space }\)
\(*V_1 \otimes ... \otimes V_k\): \(= F (V_1 \times ... \times V_k) / (S)\), the quotient vectors space
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Conditions:
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For each \((v_1, ..., v_k) \in V_1 \times ... \times V_k\), \([f_{(v_1, ..., v_k)}] = [((v_1, ..., v_k))]\) (where \(f_{(v_1, ..., v_k)}: V_1 \times ... \times V_k \to F \in F (V_1 \times ... \times V_k, F)\) is the function that maps \((v_1, ..., v_k)\) to \(1\) and maps the other elements to \(0\)) is often denoted as \(v_1 \otimes ... \otimes v_k\) and called "tensor product of \(v_1, ..., v_k\)", but one finds them misleading to be confused with tensor product of tensors and prefers denoting just \([((v_1, ..., v_k))]\): for the reason why we use the notation like \(((v_1, ..., v_k))\), see Note for the definition of free vectors space on set: \((v_1, ..., v_k) \in V_1 \times ... \times V_k\) but \(((v_1, ..., v_k)) \in F (V_1 \times ... \times V_k, F)\).
2: Note
We need to be careful not to do like "\([r ((v_1, ..., v_k)) + r' ((v'_1, ..., v'_k))] = [(r (v_1, ..., v_k)) + (r' (v'_1, ..., v'_k))] = [((r v_1, ..., r v_k)) + ((r' v'_1, ..., r' v'_k))] = [((r v_1 , ..., r v_k) + (r' v'_1, ..., r' v'_k))] = [((r v_1 + r' v'_1, ..., r v_k + r' v'_k))]\)", which is wrong: the reason is described in Note for the definition of free vectors space on set: the 1st equal is wrong because \(r ((v_1, ..., v_k)) + r' ((v'_1, ..., v'_k))\) is the function that maps \((v_1, ..., v_k)\) to \(r\) and maps \((v'_1, ..., v'_k)\) to \(r'\), while \((r (v_1, ..., v_k)) + (r' (v'_1, ..., v'_k))\) is the function that maps \(r (v_1, ..., v_k)\) to \(1\) and maps \(r' (v'_1, ..., v'_k)\) to \(1\) but maps \((v_1, ..., v_k)\) to \(0\) and maps \((v'_1, ..., v'_k)\) to \(0\); the 3rd equal is wrong because \(((r v_1, ..., r v_k)) + ((r' v'_1, ..., r' v'_k))\) is the function that maps \((r v_1, ..., r v_k)\) to \(1\) and maps \((r' v'_1, ..., r' v'_k)\) to \(1\), while \(((r v_1 , ..., r v_k) + (r' v'_1, ..., r' v'_k))\) is the function that maps \((r v_1 , ..., r v_k) + (r' v'_1, ..., r' v'_k)\) to \(1\) but maps \((r v_1, ..., r v_k)\) to \(0\) and maps \((r' v'_1, ..., r' v'_k)\) to \(0\).
These are 2 legitimate rules: 1) \([((v_1, ..., r v_j, ..., v_k))] = r [((v_1, ..., v_k))]\); 2) \([((v_1, ..., v_j + v'_j, ..., v_k))] = [((v_1, ..., v_j, ..., v_k))] + [((v_1, ..., v'_j, ..., v_k))]\).
1) is because \([((v_1, ..., r v_j, ..., v_k))] = [((v_1, ..., r v_j, ..., v_k)) - (((v_1, ..., r v_j, ..., v_k)) - r ((v_1, ..., v_k)))] = [r ((v_1, ..., v_k))] = r [((v_1, ..., v_k))]\).
2) is because \([((v_1, ..., v_j + v'_j, ..., v_k))] = [((v_1, ..., v_j + v'_j, ..., v_k)) - (((v_1, ..., v_j + v'_j, ..., v_k)) - ((v_1, ..., v_j, ..., v_k)) - ((v_1, ..., v'_j, ..., v_k)))] = [((v_1, ..., v_j, ..., v_k)) + ((v_1, ..., v'_j, ..., v_k))] = [((v_1, ..., v_j, ..., v_k))] + [((v_1, ..., v'_j, ..., v_k))]\).
