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



Dim scalars(1 To 10, 1 To 4) As Integer => {{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 n As Integer = 1 To Ubound(scalars)
    Dim As String cadena = ""
    Dim As Integer scalar
    For m As Integer = 1 To Ubound(scalars,2)
        scalar = scalars(n, m)
        If scalar <> 0 Then
            If scalar = 1 Then
                cadena &= "+e" & m
            Elseif scalar = -1 Then
                cadena &= "-e" & m
            Else
                If scalar > 0 Then
                    cadena &= Chr(43) & scalar & "*e" & m
                Else
                    cadena &= scalar & "*e" & m
                End If
            End If
        End If   
    Next m
    If cadena = "" Then cadena = "0"
    If Left(cadena, 1) = "+" Then cadena = Right(cadena, Len(cadena)-1)
    Print cadena
Next n
Sleep