Question

Linear algebra,


Can you please tell me what is the reverse of this equivalence relation?


- A is nonsingular/invertible
- A ~ I
- Ax = 0 has only the trivial solution
- Ax = b has a unique solution (x = A^-1 * b)


So the opposite would be..

- A is singular/non-invertible
- ???
- ???
- ???

Answer 1

@hartnn @Kainui

Answer 2

can someone help me please

Answer 3

@phi can you help?

Answer 4

I know an equivalence relation is reflexive, symmetric and transitive.

But what is the definition of "reverse" ?

Answer 5

if the "reverse" of A ~ I means
A not ~ I
then you would interpret that as all objects of type A intersected with all objects of type I is the empty set: \( A \ne I \)