Field Axioms -- from Wolfram MathWorld
Closure is pre-supposed and thus not included in the table.
Let together with the usual operations of matrix addition and matrix multiplication in . Prove that is a field.
A hint was provided as follows:
Prove that is closed under addition and multiplication and then go through each axiom of a field. You may simply reference well-known facts about matrix multiplication, i.e. .
This is probably easier than I think it is, but I'm still highly uncertain when it comes to fields and matrices.
Its a simple walk through axioms of field.
Comutativity, associativity are easily established. Inherited elementwise. Unique neutral element with respect to + operation: nullmatrix. Every element has a unique inverse with respect to + operation. for A in W, -A is inverse.
Commutativity easily checked, also check for associativity. Unique neutral element with respect to matrix multiplication: identity matrix.
For every element a unique inverse with respect to matrix multiplication can be found:
EDIT: I forget the distributivity. I peeked behind the link undefined posted above.