Please explain isomorphism with respect to vector spaces.

[mrroboto]

Can someone explain isomorphism to me, with respect to vector spaces. Thanks!

 

[rs1n]

Can someone explain isomorphism to me, with respect to vector spaces. Thanks!

Are you looking for a "layman's terms" explanation? You can think of an isomorphism (with respect to a vector space) as a one-to-one map from the vector space onto itself that preserves all the properties of that vector space. The image of the vector space under this map is "identical" to the original vector space. These maps are often associated with invertible matrices.

Here's an example:

Consider $$f:\mathbb{R}^2 \to \mathbb{R}^2$$ defined by

$$f(v) = v\cdot \left[ \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right]$$

where $$v$$ is a point in the plane written as row vector. That is, if $$v=(x,y)$$, then $$f(v) = f((x,y)) = (x,-y)$$. Under this map, the "y" values of the Cartesian plane are negated, while the "x" values remain the same.

 

[JasonRox]

An isomorphism from a vector space V to W is a function that is one-to-one and onto that also preserves the operation of the vector spaces.

For example, a bijective linear map is an isomorphism between vector spaces.

http://en.wikipedia.org/wiki/Isomorphism#Practical_example

 

[quasar_4]

in general, isomorphisms preserve some kind of structure. If you've heard of a homomorphism, an isomorphism is just a 1-1 and onto homomorphism.

Any linear transformation between vector spaces is actually just a homomorphism of vector spaces. When it's an invertible, 1-1, onto homomorphism (linear transformation) then it's an isomorphism.

 

[HallsofIvy]

In a very real sense, two vector spaces (or groups, or semigroups, or fields) are "isomorphic" if they are exactly the same, except the elements, operations, etc., are named differently.

If a "mathematical structure"- a set of objects $\{x_0, x_1, ...\}$ with operations {+, *, ...} is "isomorphic" to another such structure- a set $\{y_0, y_1,...\}$ with operations {+', *', ...} then there is a function, f, that maps each x to a y and each operation to a corresponding ' operation so that the operations are "preserved". If I were to 're-name' x, f(x), and rename each operation with its corresponding operation, then the "x" structure would be indistinguishable from the "y" structure.