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



function t2s(t)
    local s = "["
    for i,v in pairs(t) do
        if i > 1 then
            s = s .. ", " .. v
        else
            s = s .. v
        end
    end
    return s .. "]"
end
 
function linearCombo(c)
    local sb = ""
    for i,n in pairs(c) do
        local skip = false
 
        if n < 0 then
            if sb:len() == 0 then
                sb = sb .. "-"
            else
                sb = sb .. " - "
            end
        elseif n > 0 then
            if sb:len() ~= 0 then
                sb = sb .. " + "
            end
        else
            skip = true
        end
 
        if not skip then
            local av = math.abs(n)
            if av ~= 1 then
                sb = sb .. av .. "*"
            end
            sb = sb .. "e(" .. i .. ")"
        end
    end
    if sb:len() == 0 then
        sb = "0"
    end
    return sb
end
 
function main()
    local 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 i,c in pairs(combos) do
        local arr = t2s(c)
        print(string.format("%15s -> %s", arr, linearCombo(c)))
    end
end
 
main()