Task

Display a finite linear combination in an infinite vector basis 



(

e

1


,

e

2


,
…
)


{\displaystyle (e_{1},e_{2},\ldots )}

.

Write a function that, when given a finite list of scalars 



(

α

1


,

α

2


,
…
)


{\displaystyle (\alpha ^{1},\alpha ^{2},\ldots )}

, 
creates a string representing the linear combination 




∑

i



α

i



e

i




{\displaystyle \sum _{i}\alpha ^{i}e_{i}}

 in an explicit format often used in mathematics, that is:






α


i

1





e


i

1




±

|


α


i

2





|


e


i

2




±

|


α


i

3





|


e


i

3




±
…


{\displaystyle \alpha ^{i_{1}}e_{i_{1}}\pm |\alpha ^{i_{2}}|e_{i_{2}}\pm |\alpha ^{i_{3}}|e_{i_{3}}\pm \ldots }


where 




α


i

k




≠
0


{\displaystyle \alpha ^{i_{k}}\neq 0}







The output must comply to the following rules:

   don't show null terms, unless the whole combination is null. 
 e(1)     is fine,     e(1) + 0*e(3)     or     e(1) + 0     is wrong.
   don't show scalars when they are equal to one or minus one. 
 e(3)     is fine,     1*e(3)     is wrong.
   don't prefix by a minus sign if it follows a preceding term.   Instead you use subtraction. 
 e(4) - e(5)     is fine,     e(4) + -e(5)     is wrong.


Show here output for the following lists of scalars:

 1)    1,  2,  3
 2)    0,  1,  2,  3
 3)    1,  0,  3,  4
 4)    1,  2,  0
 5)    0,  0,  0
 6)    0
 7)    1,  1,  1
 8)   -1, -1, -1
 9)   -1, -2,  0, -3
10)   -1



#lang racket/base
(require racket/match racket/string)
 
(define (linear-combination->string es)
  (let inr ((es es) (i 1) (rv ""))
    (match* (es rv)
      [((list) "") "0"]
      [((list) rv) rv]
      [((list (? zero?) t ...) rv)
       (inr t (add1 i) rv)]
      [((list n t ...) rv)
       (define ±n
         (match* (n rv)
           ;; zero is handled above
           [(1 "") ""]
           [(1 _) "+"]
           [(-1 _) "-"]
           [((? positive? n) (not "")) (format "+~a*" n)]
           [(n _) (format "~a*" n)]))
       (inr t (add1 i) (string-append rv ±n "e("(number->string i)")"))])))
 
(for-each
 (compose displayln linear-combination->string)
 '((1 2 3)
   (0 1 2 3)
   (1 0 3 4)
   (1 2 0)
   (0 0 0)
   (0)
   (1 1 1)
   (-1 -1 -1)
   (-1 -2 0 -3)
   (-1)))
 