u/Aggressive-Food-1952

In abstract algebra we learned that, say, if two objects are isomorphic, they are essentially the same and all (I think) properties are translated. But when is this not true? Are there general properties that are not preserved or does the definition of an isomorphism automatically force it to hold?

I ask because we learned that V^n is isom. to R^n (which I think is so cool that you can just study the plane as vectors and properties of spaces).

Following this, do the normal notions of vector spaces hold in R^n? Like V^3 and R^3, clearly the dimension is the dimension, but what do the other things correspond to? Like a linear transformation or a basis or subspaces or dot products, so many things!!

I don’t understand it. The basic definitions don’t satisfy me—they seem like a massive oversimplification (but I could be wrong and overthinking it).

Can someone help me understand it? I have strong backgrounds in calculus, linear algebra, statistics and probability, and theoretical mathematics, if that helps.

I recently stumbled across the fact that -i = 1/i… is this true for any other numbers?

If such an equality exists, wouldn’t that cause problems? I’ve never taken a complex analysis class, but it’s such a unique identity that it has me questioning what its uses could be.

A little embarrassing to admit, but I honestly never understood induction. Sure I can apply it, but it just seems illogical to me. I’m wondering if anyone can clear it up for me.

Let S(n) be the inductive hypothesis. We assume it’s true to begin with, then we try to show that S(n) implies S(n+1). But this doesn’t make sense to me. Are we not assuming that the statement is valid to begin with if we assume S(n)? Isn’t it circular reasoning? I’ve heard induction has something to do with the definition of the natural numbers and the successor function. What is this about?

In a field, all nonzero elements have a multiplicative inverse... but why? Why doesn’t 0 have an inverse?

Sure, we can’t divide by 0, but in an abstract sense, we invented 0 as an element of the field. In any arbitrary field, “0” as a number might not even exist since we don’t know what these elements are!

So without saying “dividing by 0 is illegal,” is there a formal reason why we don’t have an inverse for 0? Is it just due to convention?