# Are these vector spaces isomorphs?

• gotmejerry
In summary, an isomorphism is a bijective linear transformation between two vector spaces that preserves the algebraic structure. To prove that two vector spaces are isomorphic, you must show that there exists a bijective linear transformation between them. Two vector spaces of different dimensions cannot be isomorphic, as they must have the same dimension. All subspaces of a vector space are isomorphic to the original space, and isomorphisms are significant in mathematics as they allow for the study and comparison of complex structures.

#### gotmejerry

Let K1={a + (2^0.5)*b} | a,b rational numbers}, and K2={a + (3^0.5)*b} | a,b rational numbers} be two fields with the common multiplication and addition. Isomorphs are the following vector spaces :
(Q^n ., +; K1) and (Q^n ., +; K2) ?

two finite dimensional vector spaces over the same underlying field are isomorphic if and only if they have the same dimension.

## 1. What is an isomorphism?

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.

## 2. How do you prove that two vector spaces are isomorphic?

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.

## 3. Can two vector spaces of different dimensions be isomorphic?

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.

## 4. Are all subspaces of a vector space isomorphic to the original space?

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.

## 5. What is the significance of isomorphisms in mathematics?

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.