What is an Isomorphism: Linear algebra [duplicate]

This is a rather soft question to I will tag it as such.

Basically what I am asking, is if anyone has a good explanation of what a homomorphism is and what an isomorphism is, and if possible specifically pertaining to beginner linear algebra.

This is because, in my courses we have talked about vector spaces, linear transformations, etc., but we have always for some reason skipped the sections on isomorphisms and homomorphisms.

And yes I have tried to look on wikipedia and such, but it just isn't really clicking for me what it is and what it represents/use of it.

I am under the impression that two spaces with bijection are isomorphic to one another, but that is about it.

Any ideas/opinions?

Thanks!

Solution 1:

A homomorphism is a structure-preserving mapping.
An isomorphism is a bijective homomorphism.

"Structure" can mean many different things, but in the context of linear algebra, almost exclusively means the vectorial structure -- i.e. all those rules about addition and scalar multiplication. So a homomorphism is just a mapping from one vector space to another vector space that "preserves addition and scalar multiplication". I.e. a homomorphism in linear algebra is just a linear transformation. A typical linear transformation in linear algebra is $\mathbf x \mapsto A\mathbf x$, where $\mathbf x \in \Bbb R^m$ and $A$ is an $n\times m$ matrix.

An isomorphism is not only a homomorphism -- it's bijective. So these are the invertible linear transformations. So the mapping $\mathbf x \mapsto A\mathbf x$ is an isomorphism if $A$ is an invertible matrix.

Solution 2:

Well the standard answer to this sort of question is that two algebraic objects (vector spaces in this case) $V$ and $W$ are isomorphic if they are basically the same, meaning that once can identify them with one another in a reasonable way. Another way to say this is that a map $f:V\to W$ is an isomorphism if it is bijective and it preserves the algebraic structures of $V$ (or $W$, since the definition implies that $f^{-1}:W\to V$ is an isomorphism).

In the case of vector spaces, the algebraic structure we're interested in is addition of vectors and multiplication by scalars, so that's the basis (no pun intended) for the definition $f(av_1+bv_2)=af(v_1)+bf(v_2)$ in the definition of a homomorphism (linear transformation) of vector spaces).

Why did I say basically the same and not exactly the same? The typical example here would be $V=\mathbb{R}^2$ and $W=\mathbb{C}$ (over $\mathbb{R}$). These two vector spaces aren't exactly the same, since $\mathbb{C}$ has a lot of different algebraic properties than $\mathbb{R}^2$. However, as vector spaces, the map taking $(1,0)\mapsto 1$ and $(0,1)\mapsto i$ is an isomorphism (as you can check), so as real vector spaces, they are essentially the same.

Another note here is that we used a specific choice of basis in this example for the isomorphism. There are spaces which are canonically isomorphic, which essentially means we can create an isomorphism between them that doesn't depend on the choice of basis. I can't think of an elementary linear algebra example right now other than $V$ and $V^{\ast\ast}$, the double dual space, are naturally isomorphic when $V$ is finite dimensional. Still, while basically the same, they are not exactly the same: one is a vector space of just abstract vectors, while the other is a space of functions from $V^{\ast}\to \mathbb{F}$ (the base field).

Hope that helps.

Solution 3:

Two vector spaces are isomorphic if they have the same dimension. This is equivalent to the existence of a bijective linear mapping between the spaces since to say $dim(V)=n$ is to say $\beta = \{ v_1, \dots ,v_n \}$ is a basis for $V$ and to say $dim(W)=n$ is to say $\gamma = \{ w_1, \dots ,w_n \}$ is a basis for $W$. Then we can define a linear transformation by simply setting: $$ T(v_1)=w_1, \ \ T(v_2)=w_2, \ \ \dots \ \ T(v_n)=w_n $$ then extend linearly. This makes $T: V \rightarrow W$ an invertible linear map. In short, $V$ and $W$ have the same vector space structure

The term homomorphism is naturally identified with linear transformation in this context as a linear transformation preserves the linear structure of a vector space. However, linear transformations need not be injective or surjective hence a linear transformation may or may not be an isomorphism.

The more general use of the term homomorphism or isomorphism depends on context. For groups preserve group structure. For Lie algebra, you get a vector space isomorphism which also preserves a Lie bracket. There are many many more cases.