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



with javascript_semantics
function linear_combination(sequence f)
    string res = ""
    for e=1 to length(f) do
        integer fe = f[e]
        if fe!=0 then
            if fe=1 then
                if length(res) then res &= "+" end if
            elsif fe=-1 then
                res &= "-"
            elsif fe>0 and length(res) then
                res &= sprintf("+%d*",fe)
            else
                res &= sprintf("%d*",fe)
            end if
            res &= sprintf("e(%d)",e)
        end if
    end for
    if res="" then res = "0" end if
    return res
end function
 
constant tests = {{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 i=1 to length(tests) do
    sequence ti = tests[i]
    printf(1,"%12s -> %s\n",{sprint(ti), linear_combination(ti)})
end for
