- #1

- 14

- 0

## Homework Statement

Why, in this lemma, must there be a vector v in V? That is, why must V be nonempty?

An isomorphism maps a zero vector to a zero vector.

Where f:V->W is an isomorphism, fix any vector v in V. Then f(0 vector represented with respect to V) = f(0 * vector v) = 0 * f(vector v) = 0 vector represented with respect to W.

## Homework Equations

The answer is given as "No vector space has the empty set underlying it. We can take vector v to be the zero vector."

## The Attempt at a Solution

So actually, I'm not trying to solve the problem. I'm just having a hard time understanding the answer.

What does it mean by "no vector space has the empty set underlying it?" Does that mean no vector space consists entirely of the empty set? The way it's phrased makes it sound like the vector space can include the empty set along with other sets. Wouldn't you be able to take vector v to be the zero vector in either case? Or is there no zero vector for the empty set? Even if you couldn't, why would you need to be able to take vector v to be the zero vector? A scalar zero times anything should be the zero vector, right? Or am I misinterpreting that, in that you're not supposed to take v as the zero vector?

I haven't studied in weeks, so these are possibly/probably stupid questions, I feel like I've forgotten all the basic math I learned when I started linear algebra.