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This should just about do it. Most readers still won't know why all this is true.
:Each index cannot appear more than twice in any term, so <math>a_{ij}=b_{ik}c_{kl}d_{lj}</math> is valid but <math>a_{ij}=b_{ik}c_{kk}d_{kj}</math> is not as it has more than two k's.
:Each index cannot appear more than twice in any term, so <math>a_{ij}=b_{ik}c_{kl}d_{lj}</math> is valid but <math>a_{ij}=b_{ik}c_{kk}d_{kj}</math> is not as it has more than two k's.
:Each term must contain the same non-repeated indices, so <math>a_{ij}x_j + b_{ij}x_j</math> is valid but <math>a_{ij}x_j + b_{lj}x_j</math> is not as the first term does not have the index l.
:Each term must contain the same non-repeated indices, so <math>a_{ij}x_j + b_{ij}x_j</math> is valid but <math>a_{ij}x_j + b_{lj}x_j</math> is not as the first term does not have the index l.
::{| cellspacing="0" cellpadding="0" style="margin:2em 2em 1em 0em; width:80%"
| style="vertical-align:top; border:1px solid Indigo; background-color:Lavender;" |
<div style="border-bottom:1px solid Indigo; background-color:#c0c0fa; padding:0.0em 0.5em 0.0em 0.5em; font-size:110%; font-weight:bold;">'''REALLY CORRECT TENSOR INDEX NOTATION'''</div>
<div style="padding:0.4em 1em 0.3em 1em;">
The rules for really correct tensor notation, including Einstein Summation, are as follows:
:Einstein summation must be over one subscript and one superscript.
:Indices that appear in terms to be added or subtracted, or on opposite sides of and equals sign, must match: subscript for subscript, and superscript for superscript.
:For any index left over after summation, there is an implicit "for all" in the equation.
So the above equation becomes just
:<math>
b^i = a^i_j x^j
</math>
That is the really proper way to express that vector <math>\vec{\boldsymbol{b}}</math> is the result of applying the linear transformation <math>\boldsymbol{A}</math> to the vector <math>\vec{\boldsymbol{x}}</math>
</div>
|}
In general, the indices may subscript or superscript depending on whether the tensors are covariant or contravariant, respectively.
In general, the indices are subscript or superscript depending on whether the tensors are covariant or contravariant, respectively.
==References==
==References==