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10. Classification (with Selection and Inner Product)
The complete classification table can be used in a variety of ways, including use with various inner products. For example:

      cct=: #:@i.@(2: ^ #)
      m=: 2 3 5 ,: 4 2 1
      n=: |: cct 0{m
      m ; n ; m +/ . * n
    +-----+---------------+----------------+
    |2 3 5|0 0 0 0 1 1 1 1|0 5 3 8 2 7 5 10|
    |4 2 1|0 0 1 1 0 0 1 1|0 1 2 3 4 5 6  7|
    |     |0 1 0 1 0 1 0 1|                |
    +-----+---------------+----------------+

The pattern of the inner product can be seen more clearly in the following display: the element in a given row and column of the matrix product`p` in the lower right box corresponds to the row of the left argument and the column of the right argument. Thus:

       ('' ; n) ,: (m ; p=: m +/ . * n)
    +-----+----------------+
    |     |0 0 0 0 1 1 1 1 |
    |     |0 0 1 1 0 0 1 1 |
    |     |0 1 0 1 0 1 0 1 |
    +-----+----------------+
    |2 3 5|0 5 3 8 2 7 5 10|
    |4 2 1|0 1 2 3 4 5 6  7|
    +-----+----------------+

       (+/r*c) ; (r*c) ; (r=: 0{m) ; (c=: 3{"1 n) ; (<0 3){p
    +-+-----+-----+-----+-+
    |8|0 3 5|2 3 5|0 1 1|8|
    +-+-----+-----+-----+-+

Just as the ordinary matrix product yields sums over products, the inner product`*/ . ^` yields products over powers. Hence,`m */ . ^ n` produces products over all possible subsets of the rows of`m`:

       m */ . ^ n
    1 5 3 15 2 10 6 30
    1 1 2  2 4  4 8  8

Also see the use of the key adverb (`/.`) for classification.

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