# Books/MathForTheLayman/Linear Vector Functions

# 15: Linear Vector Functions

## 15A. Dot product and vector functions

The expression +/w*x gives a sum over the argument x weighted by the vector w. For example:

x=: 2 3 5 7 11 w=: 0 1 2 3 4 w*x 0 3 10 21 44 +/w*x 78

The dot operator (.) applied to the functions +/ and * produces an equivalent function called the dot, inner, or matrix product. Thus:

dp=: +/ . * w dp x 78

Moreover, dp applies to a matrix (table) left argument to produce a vector (list) of results, using successive rows of the matrix as weights. For example:

a=: 3 1 4 b=: 2 7 8 c=: 2 1 0 ]m=: a,b,:c 3 1 4 2 7 8 2 1 0

m dp 2 3 5 29 65 7 b dp 2 3 5 65

g=: m with dp g 2 3 5 29 65 7

If m is a square matrix (as in the present example), then m with dp applies to a vector to produce a vector of the same number of elements. We will call such a function a vector function.

## 15B. Dot product as a linear vector function

The dot product with a vector or matrix left argument distributes over addition, and is therefore a linear function, as defined in Section 14F. For example:

x=: 2 3 5 y=: 0 1 2 f=: 2 7 8 with dp

((f x)+(f y)) , (f x+y) 88 88 g=: m with dp ((g x)+(g y)) ,: (g x+y) 38 88 8 38 88 8

The function I=: =/~ on i. yields a result called an identity matrix, because when used with dp it produces an identity function (that yields its argument unchanged). For example:

I=: i. =/ i. ]i=: I 3 1 0 0 0 1 0 0 0 1 i with dp 2 3 5 2 3 5

A linear vector function need not be expressed as a dot product, but if it is not, the required matrix is easily determined. For example, sop=: +/\ (sums over prefixes) is a linear vector function, and the matrix m for an equivalent function m with dp may be determined as follows:

x=: 2 3 5 7 11 y=: 0 1 2 3 4 sop=: +/\ ((sop x)+(sop y)) ,: (sop x+y) 2 6 13 23 38 2 6 13 23 38

]i=: I # x 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

i dp x 2 3 5 7 11 sop i dp x 2 5 10 17 28 (sop i) dp x 2 5 10 17 28

]m=: sop i 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 1 1 1 1

The resulting matrix provides an interesting way of looking at the sums over prefixes: the ones in the successive rows indicate the elements included in the successive sums. 15C. Matrix inverse S15C. The inverse of a linear vector function m with dp is itself a linear vector function, and can therefore be expressed in the form mi with dp. The matrix mi may be obtained from m by applying the matrix inverse function %. . Thus:

]z=: m dp x=: 2 3 5 7 11 2 5 10 17 28 mi=: %.m mi dp z 2 3 5 7 11 mi 1 0 0 0 0 _1 1 0 0 0 0 _1 1 0 0 0 0 _1 1 0 0 0 0 _1 1

The pairs _1 1 in the rows of the inverse mi indicate that the inverse of the sum over prefixes is a differencing operation, and the matrix mi may be called a difference matrix.

**Exercises**

Experiment with various linear vector functions such as reversal (|.), rotation (2 with |.) and 8 with A..

Comment on the matrices that represent them and their inverses.