Help / Phrases / 9. B. Linear Vector Functions

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9B. Linear Vector Functions

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 is a test for the linearity of f:

   f=: +/\"1	  NB. 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@:+) 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	    NB. 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	    NB. Identity matrix
   mf=: (f I) [ mg=: (g I)
   mp=: +/ . *	    NB. 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=: ]:@=@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=: inv '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 3  5  7 11│
│0 0 0 1 1│0  0  0  1 _1│            │
│0 0 0 0 1│0  0  0  0  1│            │
└─────────┴─────────────┴────────────┘

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