terminology - What does isomorphic mean in linear algebra . . . I've also heard that this is an abstract algebra term, so I'm not sure if isomorphic means the same thing in both subjects, but I know absolutely no abstract algebra, so in your definition if you keep either keep abstract algebra out completely, or use very basic abstract algebra knowledge, that would be appreciated
what exactly is an isomorphism? - Mathematics Stack Exchange An isomorphism is a particular type of map, and we often use the symbol $\cong$ to denote that two objects are isomorphic to one another Two objects are isomorphic there is a $1$-$1$ map from one object onto the other that preserves all of the structure that we're studying That second part is important, but it's often implied from context
abstract algebra - What is exactly the meaning of being isomorphic . . . $\begingroup$ The example of geometry is badly chosen There are many different forms of geometry (Euclidean, projective, Riemannian, differential, to name a few) which are totally incomparable; not only they are not isomorphic, they don't even have the same language, so talking about isomorphism between then is meaningless
What does it mean when two Groups are isomorphic? For sets: isomorphic means same cardinality, so cardinality is the "classifier" For vector spaces: isomorphic means same dimension, so dimension (i e , cardinality of a base) is our classifier I is a bit more complex but still not too difficult (you'll probably encounter it in your book sooner or later) to classify finite abelian groups
How to tell whether two graphs are isomorphic? Importantly, it does not tell us that the two other graphs are isomorphic, even though they have the same degree sequence In fact, they are not isomorphic either: in the middle graph, the unique vertex of degree $5$ is adjacent to a vertex of degree $2$ , and in the graph on the right, the unique vertex of degree $5$ is only adjacent to
Whats the difference between isomorphism and homeomorphism? What does it mean when two things are homomorphic (its clear that when two things are isomorphic, they are essentially the same object with different labels)? $\endgroup$ – Madhav Nakar Commented Dec 6, 2019 at 21:16
Are these two graphs isomorphic? Why Why not? Two things are isomorphic given an isomorphism, but you don't give one Lacking one, common sense suggests "isomorphic" means for some isomorphism of a given kind For graphs "isomorphic" assumes a certain kind of isomorphism You are misusing descriptions that are too vague to be definitions
Difference between ≈, ≃, and ≅ - Mathematics Stack Exchange For example "geometrically isomorphic" usually means "congruent," "topologically isomorphic" means "homeomorphic," et cetera: it means they're somehow the "same" for the structure you're considering, in some senses they are "equivalent," though not always "equal:" you could have two congruent triangles at different places in a plane, so they
Are the groups $(\\mathbb{C},+)$ and $(\\mathbb{R},+)$ isomorphic? Therefore their additive groups are isomorphic $\mathbb{Q}$ is a one-dimensional $\mathbb{Q}$-vector space whereas $\mathbb{Q}[i]$ is a two-dimensional $\mathbb{Q}$-vector space, so their additive groups are not isomorphic (Note that the dimension of a $\mathbb{Q}$-vector space is the maximum cardinality of a $\mathbb{Z}$-linearly independent