description/proof of that for tensors space w.r.t. field and \(k\) finite-dimensional vectors spaces over field and field, transition of components of tensor w.r.t. standard bases w.r.t. bases for vectors spaces is this
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of tensors space with respect to field and k vectors spaces and vectors space over field.
- The reader admits the proposition that for the tensors space with respect to any field and any \(k\) finite-dimensional vectors spaces over the field and the field, the transition of the standard bases with respect to any bases for the vectors spaces is this.
- The reader admits the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this.
- The reader admits the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses.
Target Context
- The reader will have a description and a proof of the proposition that for the tensors space with respect to any field and any \(k\) finite-dimensional vectors spaces over the field and the field, the transition of the components of any tensor with respect to the standard bases w.r.t. any bases for the vectors spaces is this.
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:
\(F\): \(\in \{\text{ the fields }\}\)
\(\{V_1, ..., V_k\}\): \(\subseteq \{\text{ the finite-dimensional } F \text{ vectors spaces }\}\)
\(L (V_1, ..., V_k: F)\): \(= \text{ the tensors space }\)
\(\{B_1, ..., B_k\}\): \(B_j \in \{\text{ the bases for } V_j\} = \{{b_j}_l \vert 1 \le l \le dim V_j\}\)
\(\{B'_1, ..., B'_k\}\): \(B'_j \in \{\text{ the bases for } V_j\} = \{{b_j}_l \vert 1 \le l \le dim V_j\}\)
\(\{B^*_1, ..., B^*_k\}\): \(B^*_j = \text{ the dual basis of } B_j = \{{b_j}^l \vert 1 \le l \le dim V_j\}\)
\(\{B'^*_1, ..., B'^*_k\}\): \(B'^*_j = \text{ the dual basis of } B_j = \{{b_j}^l \vert 1 \le l \le dim V_j\}\)
\(B^*\): \(= \{{b_1}^{j_1} \otimes ... \otimes {b_k}^{j_k} \vert {b_l}^{j_l} \in B^*_l\}\), \(\in \{\text{ the bases for } L (V_1, ..., V_k: F)\}\)
\(B'^*\): \(= \{{b'_1}^{j_1} \otimes ... \otimes {b'_k}^{j_k} \vert {b'_l}^{j_l} \in B'^*_l\}\), \(\in \{\text{ the bases for } L (V_1, ..., V_k: F)\}\)
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Statements:
\({b'_j}_l = {b_j}_m {M_j}^m_l\)
\(\implies\)
\(\forall f = f_{j_1, ..., j_k} {b_1}^{j_1} \otimes ... \otimes {b_k}^{j_k} = f'_{l_1, ..., l_k} {b'_1}^{l_1} \otimes ... \otimes {b'_k}^{j_l} \in L (V_1, ..., V_k: F) (f'_{l_1, ..., l_k} = f_{j_1, ..., j_k} {M_1}^{j_1}_{l_1} ... {M_k}^{j_k}_{l_k})\)
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2: Proof
Whole Strategy: just apply the proposition that for the tensors space with respect to any field and any \(k\) finite-dimensional vectors spaces over the field and the field, the transition of the standard bases with respect to any bases for the vectors spaces is this, the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this, and the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses; Step 1: see that \({b'_1}^{j_1} \otimes ... \otimes {b'_k}^{j_k} = {{M_1}^{-1}}^{j_1}_{l_1} ... {{M_k}^{-1}}^{j_k}_{l_k} {b_1}^{l_1} \otimes ... \otimes {b_k}^{l_k}\); Step 2: conclude the proposition.
Step 1:
\({b'_1}^{j_1} \otimes ... \otimes {b'_k}^{j_k} = {{M_1}^{-1}}^{j_1}_{l_1} ... {{M_k}^{-1}}^{j_k}_{l_k} {b_1}^{l_1} \otimes ... \otimes {b_k}^{l_k}\), by the proposition that for the tensors space with respect to any field and any \(k\) finite-dimensional vectors spaces over the field and the field, the transition of the standard bases with respect to any bases for the vectors spaces is this.
Step 2:
Let us apply the proposition that for any finite-dimensional vectors space, the transition of the components of any vector with respect to any change of bases is this.
By the proposition that for the tensors space with respect to any field and any finite number of finite-dimensional the field vectors spaces and the field or the tensor product of any finite-dimensional vectors spaces over any field, the transition of any standard bases or the components is a square matrix, and the inverse matrix is the product of the inverses, \(f'_{l_1, ..., l_k} = f_{j_1, ..., j_k} {M_1}^{j_1}_{l_1} ... {M_k}^{j_k}_{l_k}\).
