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



using System;
using System.Collections.Generic;
using System.Text;
 
namespace DisplayLinearCombination {
    class Program {
        static string LinearCombo(List<int> c) {
            StringBuilder sb = new StringBuilder();
            for (int i = 0; i < c.Count; i++) {
                int n = c[i];
                if (n < 0) {
                    if (sb.Length == 0) {
                        sb.Append('-');
                    } else {
                        sb.Append(" - ");
                    }
                } else if (n > 0) {
                    if (sb.Length != 0) {
                        sb.Append(" + ");
                    }
                } else {
                    continue;
                }
 
                int av = Math.Abs(n);
                if (av != 1) {
                    sb.AppendFormat("{0}*", av);
                }
                sb.AppendFormat("e({0})", i + 1);
            }
            if (sb.Length == 0) {
                sb.Append('0');
            }
            return sb.ToString();
        }
 
        static void Main(string[] args) {
            List<List<int>> combos = new List<List<int>>{
                new List<int> { 1, 2, 3},
                new List<int> { 0, 1, 2, 3},
                new List<int> { 1, 0, 3, 4},
                new List<int> { 1, 2, 0},
                new List<int> { 0, 0, 0},
                new List<int> { 0},
                new List<int> { 1, 1, 1},
                new List<int> { -1, -1, -1},
                new List<int> { -1, -2, 0, -3},
                new List<int> { -1},
            };
 
            foreach (List<int> c in combos) {
                var arr = "[" + string.Join(", ", c) + "]";
                Console.WriteLine("{0,15} -> {1}", arr, LinearCombo(c));
            }
        }
    }
}