An isomorphism is a bijective linear transformation between two vector spaces that preserves the algebraic structure. In other words, it is a one-to-one and onto mapping that preserves the operations of addition and scalar multiplication.
To prove that two vector spaces are isomorphic, you must show that there exists a bijective linear transformation between them. This can be done by showing that the two vector spaces have the same dimension, and then constructing a specific linear transformation that maps one basis to the other.
No, two vector spaces of different dimensions cannot be isomorphic. Isomorphic vector spaces must have the same dimension, as this is a necessary condition for a bijective linear transformation to exist between them.
Yes, all subspaces of a vector space are isomorphic to the original space. This is because subspaces inherit the same algebraic structure as the original space, and thus, the same linear transformation that maps the original space to itself can also map the subspace to itself.
Isomorphisms are important in mathematics because they allow us to study and understand complex structures by breaking them down into simpler, isomorphic structures. They also provide a way to compare and relate different mathematical objects, and can help us identify similarities and differences between them.