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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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