# remainder' -- matrix quotient and remainder (opposite)

## Synopsis

• Usage:
(q,r) = remainder'(f,g)
• Inputs:
• f,
• g, or , with the same source as f
• Outputs:
• r, the remainder of f upon (opposite) division by g

## Description

The equation q*g+r == f will hold, where q is the map provided by quotient'. The sources and targets of the maps should be free modules. This function is obtained from remainder by transposing the inputs and outputs.
 ```i1 : R = ZZ[x,y] o1 = R o1 : PolynomialRing``` ```i2 : f = random(R^{2:1},R^2) o2 = {-1} | 8x x+5y | {-1} | 6x+9y 6x+8y | 2 2 o2 : Matrix R <--- R``` ```i3 : g = transpose (vars R ++ vars R) o3 = {-1} | x 0 | {-1} | y 0 | {-1} | 0 x | {-1} | 0 y | 4 2 o3 : Matrix R <--- R``` ```i4 : remainder'(f,g) o4 = 0 2 2 o4 : Matrix R <--- R``` ```i5 : f = f + map(target f, source f, id_(R^2)) o5 = {-1} | 8x+1 x+5y | {-1} | 6x+9y 6x+8y+1 | 2 2 o5 : Matrix R <--- R``` ```i6 : remainder'(f,g) o6 = {-1} | 1 0 | {-1} | 0 1 | 2 2 o6 : Matrix R <--- R```

`function 'remainder'': source code not available`