micromass said:An isomorphism between which structure?? Vector spaces??
Anyway, given a symmetric matrix
[tex]\left(\begin{array}{cc} a & b\\ b & c \end{array}\right)[/tex]
what element of [itex]\mathbb{R}^3[/itex] would you associate with this matrix??
retracell said:Yes vector spaces. What do you mean by what element? Would R^3 simply be some vector v=(v1, v2, v3)?
An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures. It essentially means that two structures are identical in their underlying structure, even if their elements may be represented differently.
Constructing isomorphisms is important because it allows us to identify and compare different mathematical structures that may appear different at first glance. By finding isomorphisms, we can establish connections between seemingly unrelated structures and better understand their properties and relationships.
To construct an isomorphism between two structures, you need to find a one-to-one mapping between their elements that preserves the structure of the two structures. This means that the mapping must preserve the relationships between elements and operations in both structures.
Yes, there can be multiple isomorphisms between two structures. In fact, there can be an infinite number of isomorphisms between two structures, as long as they preserve the structure of the two structures.
No, not all structures are isomorphic to each other. In order for two structures to be isomorphic, they must have the same underlying structure and be able to be mapped onto each other in a one-to-one manner. If the structures have different underlying structures, they cannot be isomorphic.