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25. Linear Functions

A function f is said to be linear if f(x+y) equals (f x)+(f y) for all arguments x and y . For example:
```   f=: 3&|. @: +: @: |.
]x=: i.# y=:2 3 5 7 11
0 1 2 3 4

x+y                            f x+y
2 4 7 10 15                    8 4 30 20 14

(f x),:(f y)                   (f x)+(f y)
2 0  8  6  4                   8 4 30 20 14
6 4 22 14 10
```
A linear function can be defined equivalently as follows: f is linear if f@:+ and +&f are equivalent. For example:
```   x f@:+ y                       x +&f y
8 4 30 20 14                   8 4 30 20 14
```
If f is a linear function, then f y can be expressed as the matrix product mp&M y , where
```   mp=: +/ . *
M=: f I=: = i.#y            I is an identity matrix

mp&M y
6 4 22 14 10
f y
6 4 22 14 10
```
Conversely, if m is any square matrix of order #y , then m&mp is a linear function on y , and if m is invertible, then (%.m)&mp is its inverse:

```   x=: 1 2 3 [ y=: 2 3 5
]m=: ?. 3 3\$9
3 8 8
4 2 0
2 7 4
]n=: %. m
0.0909091   0.272727 _0.181818
_0.181818 _0.0454545  0.363636
0.272727 _0.0568182 _0.295455

g=: mp&m
h=: mp&n

x g@:+ y
45 90 56

x +&g y
45 90 56
g h y
2 3 5
```

Exercises

 25.1 For each of the following functions, determine the matrix M such that M (mp=: +/ . *) N is equivalent to the result of the function applied to the matrix N , and test it for the case N=: i. 6 6 ```|. - +: (4&*-2&*@|.) 2&A. ```

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