What does "$\;0'\;$" mean in "Suppose $0$ and $0'$ are both additive identities for some vector space $V$"?

Question

This is out of Sheldon Axler's Linear Algebra Done Right, page 14:

Suppose $0$ and $0’$ are both additive identities for some vector space $V$. $$0’ = 0’ + 0 = 0 + 0’ = 0$$

I’m not sure what the book means by zero prime.

Answer 1

It means a second additive identity i.e. another number such that any element $v \in V$ has the property $o' + v = v + o' = v$ as well as the regular additive identity $0 +v = v+0 = v$.

Answer 2

This is a standard technique usually used to prove that an object, if exists, is unique: if $a$ and $b$ are both that "something", then some reasoning based on the characteristics of that "something" would lead to the conclusion that $a = b$.

Answer 3

The book is just using $0'$ as a variable name, not as anything with a formal meaning. The implication of this variable name is that this new variable is related to $0$ - and, in context, the name emphasizes that $0'$ is an additive identity. The book could have just as well said:

Let $x,y\in V$ be additive identities, meaning that for all $z\in V$ we have $x+z=z+x=z$ and $y+z=z+y=z$. Then, $$x=x+y=y+x=y.$$

The conclusion, of course, is that $0'=0$, which largely justifies the naming.

Answer 4

An additive identity is a element, $a$ so that $a +V = V+a=V$ for all $V$.

The claim is that there is such an element. We give it a name. We decided to give it the name "$0$".

But we only claimed there was one such element. Can there were two or more?

What if a friennd says: Here I want you to meet someone: I call him ZERO. He's got the ability that $someone = someone + him = him + someone$; he's an additive identity.

Then at another party: A coworker says, Let me introduce you to a guy: I call him ZILCH. He's an additive identity. He's got to special quality that if you take anyone then $him+ anyone = anyone +him = anyone$. Isn't that neat?

And you say: Hey, I met an additive identity a while back named ZERO; was that you? And ZILCH says maybe, I get around a lot. Maybe it was another additive identity. There might be many of us.

And you say .... no, there's only one additive identity. I mean you are ZILCH, right. ANd since ZERO is an additive identity then ZILCH = ZILCH + ZERO, right? But you, ZILCH, are also an additive identity. So ZILCH + ZERO = ZERO. So ZILCH = ZERO. It was you. You are the only additive identity in the world. When it comes to additive identities; There Can Only Be One! (or, er, maybe I should say... There Can Only Be ZERO!)

.....

Okay.... I'm being silly.

But to your book. So what if there we imagined there were possibly two additive identities.

The book choose: Let's take an additive identity. Let's give it a name. Let's call it "$0$".

Now lets take an additive identity. It might be the same one (in fact we'll prove it must be) but (for now we'll assume) it might be a different one. Let's give it a name. We can't name it "$0$"; that name is taken. So let's name it "$0'$". The "prime" is only so we can tell these (potentially) two additive identities apart.

But as they are both additive identities we have:

$0' = 0' + 0 = 0$. SO $0$ and $0'$ must be the same thing and they are just different names for the same value. It's ZERO. The same zero we've known for years and took for granted. It's just that now, we are giving him a full examination.