directSum -- direct sum of modules or maps

Description

The components can be recovered later with components or with formation.

Projection and inclusion maps for direct sums:

• Module ^ Array -- projection onto summand
• Module _ Array -- inclusion from summand
• Matrix ^ List -- select rows
• Matrix _ List -- select columns

It sometimes happens that the user has indices for the components of a direct sum preferable to the usual consecutive small integers. In this case the preferred indices can be specified with code like directSum(a=>M,b=>N,...), as in the following example.

i1 : F = directSum(a=>ZZ^1, b=>ZZ^2, c=>ZZ^3)

       6
o1 = ZZ

o1 : ZZ-module, free

i2 : F_[b]

o2 = | 0 0 |
     | 1 0 |
     | 0 1 |
     | 0 0 |
     | 0 0 |
     | 0 0 |

              6        2
o2 : Matrix ZZ  <--- ZZ

i3 : F^[c]

o3 = | 0 0 0 1 0 0 |
     | 0 0 0 0 1 0 |
     | 0 0 0 0 0 1 |

              3        6
o3 : Matrix ZZ  <--- ZZ

i4 : F = (a => ZZ^1) ++ (b => ZZ^2)

       3
o4 = ZZ

o4 : ZZ-module, free

i5 : F_[b]

o5 = | 0 0 |
     | 1 0 |
     | 0 1 |

              3        2
o5 : Matrix ZZ  <--- ZZ

See also

• ++ -- a binary operator, usually used for direct sum
• components -- list the components of a direct sum
• indexComponents -- specify keys for components of a direct sum
• indices -- indices of a polynomial; also components for a direct sum
• formation -- recover the methods used to make a module