Metamath Proof Explorer


Table of Contents - 11.2.4. Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng.

For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric.

  1. mat0dimbas0
  2. mat0dim0
  3. mat0dimid
  4. mat0dimscm
  5. mat0dimcrng
  6. mat1dimelbas
  7. mat1dimbas
  8. mat1dim0
  9. mat1dimid
  10. mat1dimscm
  11. mat1dimmul
  12. mat1dimcrng
  13. mat1f1o
  14. mat1rhmval
  15. mat1rhmelval
  16. mat1rhmcl
  17. mat1f
  18. mat1ghm
  19. mat1mhm
  20. mat1rhm
  21. mat1rngiso
  22. mat1ric