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 10A 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 14If 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 10Conversely, 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 5 7 3 7 2 3 4 4 2 ]n=: %.m _1.33333 _0.333333 2.5 _0.333333 _0.333333 1 3.33333 1.33333 _6.5 g=: mp&m h=: mp&n x g@:+ y 82 63 40 x +&g y 82 63 40 g h y 2 3 5
Exercises
| 26.1 | 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. |