Zero vector in a vector space

Question

I know that every vector space contains a zero element so that u + 0 = 0 + u = u but I am not sure if every vector space contains the literal vector with all zero elements or if it is just a vector that acts as a zero vector?

I'm especially wondering this question because a fact about subspaces is that every vector space has at least two subspaces: itself and the zero vector {0}. But in this case, I am not sure if this refers to the vector with all zero elements or if it refers to the vector in the vector space that just acts like a zero vector

Answer 1

A vector space $V$ over the real numbers is a set $V$ together with a map $+\colon V \times V \to V$ (called "addition") and a map $\cdot\colon \mathbb{R} \times V \to V$ (called "scalar multiplication") satisfying a certain list of axioms. One of those axioms is that there is an element $0 \in V$ such that $u + 0 = 0 + u = u$ for all $u \in V$. This element is conventionally called the "zero vector" and labelled with the symbol $0$.

A key thing to note here is that this says nothing about what the elements of $V$ (either the zero vector or the other elements) are, just how they act under the given addition and scalar multiplication operations. What the elements are is completely irrelevant except to the extent that this is used in defining the vector space operations. In other words, the thing that "acts like a zero vector" is the zero vector, by definition.

In particular, there's no notion of "coordinates/entries of a vector" for elements of a general vector space. That notion arises when we choose a basis for a vector space; a choice of basis gives a one-to-one correspondence between elements of the vector space and lists of real numbers (indexed by the basis elements). In the finite-dimensional case, this gives the familiar representation of a vector as a finite list of real numbers. And once you choose a basis, you can indeed prove that the representation of the zero vector in any basis has all coordinates zero.

Answer 2

Here's a nice vector space: The set of all functions, which takes real numbers to real numbers.

This is a vector space over $\mathbb{R}$: Adding two functions together gives another function, multiplying a function with a scalar still gives a function, addition is associative, ect.

The zero vector in this example is the constant function, $z(t)=0$ for all $t$.

You can check for yourself that if you take any other function and add it to $z$, that you will get the same function back. Also, given any function $f$, you can define $g:\mathbb{R}\rightarrow\mathbb{R}$ by $g(t)=-f(t)$, and you will get that $f+g=z$. However, the function $z$ is not "the vector with all zero elements", like you would see for vectors in $\mathbb{R}^n$. In general, vector spaces do not look like lists of elements from the base field.