If f(x+y)equals (f x)+(f y) for all vectors x and y in its domain, then f is said to be a linear vector function. In other words, f@:+ -: f@[ + f@] is a test for the linearity of f: |
f=: +/\"1 Sum scan (subtotals) is a linear vector function
x=: 4 3 2 1 0
y=: 2 3 5 7 11
x ([ , ] , f@[ , f@] , (f@[ + f@]) ,: f@:+) y
4 3 2 1 0 x
2 3 5 7 11 y
4 7 9 10 10 f x
2 5 10 17 28 f y
6 12 19 27 38 (f x)+(f y)
6 12 19 27 38 f (x+y)
For vector arguments of dimension n, any linear vector function f can be expressed as a matrix product, using a matrix obtained by applying f to the identity matrix of order n. Moreover, the inverse of a linear function is a linear function. For example: |
g=: f^:_1 Inverse of subtotals is first differences
(] ; f ; g ; f@g) y
+--------------------------------------------+
�2 3 5 7 11�2 5 10 17 28�2 1 2 2 4�2 3 5 7 11�
+--------------------------------------------+
I=: = i. 5 Identity matrix
mf=: (f I) [ mg=: (g I)
mp=: +/ . * Matrix product
mf ; mg ; (y,(y mp mf),:f y) ; (mf mp mg)
+----------------------------------------------+
�1 1 1 1 1�1 _1 0 0 0�2 3 5 7 11�1 0 0 0 0�
�0 1 1 1 1�0 1 _1 0 0�2 5 10 17 28�0 1 0 0 0�
�0 0 1 1 1�0 0 1 _1 0�2 5 10 17 28�0 0 1 0 0�
�0 0 0 1 1�0 0 0 1 _1� �0 0 0 1 0�
�0 0 0 0 1�0 0 0 0 1� �0 0 0 0 1�
+----------------------------------------------+
| a0=: MR=: 1 : 'x.@=@i.@#' | Matrix representation of linear function |
| d1=: mp=: +/ . * | Matrix product |
| a2=: L=: &mp | Linear function represented by matrix |
| a3=: inv=: ^:_1 | Inverse adverb |
| a4=: MRI=: (^:_1) 'MR' f. | Matrix representation of inverse function |
| a5=: MR (%.@) | " |
|
m=: f MR y [ mi=: f MRI y
m ; mi ; y,(y L m),:(mi L m L y)
+------------------------------------+
�1 1 1 1 1�1 _1 0 0 0�2 3 5 7 11�
�0 1 1 1 1�0 1 _1 0 0�2 5 10 17 28�
�0 0 1 1 1�0 0 1 _1 0�2 5 10 17 28�
�0 0 0 1 1�0 0 0 1 _1� �
�0 0 0 0 1�0 0 0 0 1� �
+------------------------------------+
|