# quotient' -- matrix quotient (opposite)

## Synopsis

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

## Description

The equation q*g+r == f will hold, where r is the map provided by remainder' . The sources and targets of the maps should be free modules. This function is obtained from quotient 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} | x+2y 9x+4y | {-1} | 2x+8y 7x+2y | 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 : quotient'(f,g) o4 = {-1} | 1 2 9 4 | {-1} | 2 8 7 2 | 2 4 o4 : Matrix R <--- R``` ```i5 : f = f + map(target f, source f, id_(R^2)) o5 = {-1} | x+2y+1 9x+4y | {-1} | 2x+8y 7x+2y+1 | 2 2 o5 : Matrix R <--- R``` ```i6 : quotient'(f,g) o6 = {-1} | 1 2 9 4 | {-1} | 2 8 7 2 | 2 4 o6 : Matrix R <--- R```

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