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



 
 
{def linearcomb
 {def linearcomb.r
  {lambda {:a :n :i}
   {if {= :i :n}
    then 
    else {let { {:e e({+ :i 1})}
                {:v {abs {A.get :i :a}}}
                {:s {if {< {A.get :i :a} 0} then - else +}}
              } {if {= :v 0} then  else
                {if {= :v 1} then :s :e else :s :v*:e}}}
         {linearcomb.r :a :n {+ :i 1}} }}}
 {lambda {:a}
  {S.replace _LAMB_[^\s]+ by 0 in
   {let { {:r {linearcomb.r {A.new :a} {S.length :a} 0}}
        } {if {W.equal? {S.first :r} +} then {S.rest :r} else :r} }}}}
-> linearcomb
 
{linearcomb 1 2 3}      -> e(1) + 2*e(2) + 3*e(3)
{linearcomb -1 -2 0 -3} -> - e(1) - 2*e(2) - 3*e(4) 
{linearcomb 0 1 2 3}    -> e(2) + 2*e(3) + 3*e(4) 
{linearcomb 1 0 3 4}    -> e(1) + 3*e(3) + 4*e(4)
{linearcomb 1 2 0}      -> e(1) + 2*e(2) 
{linearcomb 0 0 0}      -> 0
{linearcomb 0}          -> 0 
{linearcomb 1 1 1}      -> e(1) + e(2) + e(3)  
{linearcomb -1 -1 -1}   -> - e(1) - e(2) - e(3)
{linearcomb -1}         -> - e(1)
 
 