Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm looking for a graphic that shows the isomorphism theorems illustrated using a finite vector space. (that is a finite dimensional vector space over a finite field.) Something that shows nicely how the cosets "collapse" in to single elements or how cosets at different "levels" are matched up to form the isomorphisms.

I have tried searching but I keep finding arrow diagrams.

Alternately, suggestions for good examples that don't take too long to draw would be appreciated.

share|improve this question
4  
@a little don: A simple enough picture can be done with $\mathbb{R}^2$ and $H=\{(ka,kb): x\in\mathbb{R}\}$. The cosets are the lines of slope $b/a$, and if you pick your favorite line through the original that is not of that slope, the cosets "colapse" to their projection onto that line. E.g., $H=\{(x,0):h\in \mathbb{R}\}$, and project onto the $Y$-axis to get $\mathbb{R}^2/H\cong \mathbb{R}$. –  Arturo Magidin Nov 29 '10 at 20:52

2 Answers 2

up vote 4 down vote accepted

Over 9000 hours in MSPaint and I now have the graphic Arturo constructed in his comment. Strikingly, it's also what I've always pictured in my head when I visualize cosets and quotients:

enter image description here

Each colored line is a copy of $H$ translated, and they are each individually the coset of the bold color dot where it intersects the gray line; you can see quite vividly how each colored line "collapses" into a dot on the gray one. If you haven't figure it out already, the gray line is $\mathbb{R}^2/H$.

As for the full isomorphism theorems, actually illustrating them rather than simply diagramming them should prove to be a difficult but rather rewarding project. I hope to see some talented artist take up the challenge in the future.

share|improve this answer

This is for $\mathbb{R}^n$, though it applies to $\mathbb{F}_p^n$ just as well since it is more of a metaphorical picture (e.g., the "orthogonality" of the subspaces). enter image description here

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.