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



import strformat
 
proc linearCombo(c: openArray[int]): string =
 
  for i, n in c:
    if n == 0: continue
    let op = if n < 0:
               if result.len == 0: "-" else: " - "
             else:
               if n > 0 and result.len == 0: "" else: " + "
    let av = abs(n)
    let coeff = if av == 1: "" else: $av & '*'
    result &= fmt"{op}{coeff}e({i + 1})"
  if result.len == 0:
    result = "0"
 
const Combos = [@[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]]
 
for c in Combos:
  echo fmt"{($c)[1..^1]:15}  →  {linearCombo(c)}"