I'm learning now about vector spaces and subspaces, and one of three rules that determine if something is a subspace of a larger vector space is that it must contain the zero vector… but intuitively, I can't figure out why the zero vector wouldn't exist. When I first learned about vectors over the summer, the zero vector was basically described as "pick a point on the x-y plane. Move 0 units up and 0 units to the right. That is the zero vector… just a dot on the x-y plane". This is assuming of course that you're talking about a vector $\vec{v}\in\mathbb{R}^2$.
Now fast forward to today when I'm given the following definition and need to determine if it's a subspace or not based on the three rules:
- Does it contain the zero vector?
- Is the set closed under vector addition?
- Is the set closed under scalar multiplication?
Take for example the question:
Let $a,b,c,d$ be constants, and let $U=\left\{\left[\begin{array}{r}x\\y\\z\end{array}\right]\in\mathbb{R}^{3}\;\middle|\; ax+by+cz=d\right\}$. Show that $U$ is a subspace of $V$ if and only if $d=0$.
By the first test above, $\left[\begin{array}{r}0\\0\\0\end{array}\right]$ is not in $U$ if $d\neq 0$ because $a(0)+b(0)+c(0)$ necessarily implies that $d=0$…
Consider also the line $x+y=1$ in $\mathbb{R}^2$. This also does not contain the zero vector.
It seems to me like "zero vector" is being used synonymously with "origin", but this doesn't fit the definition of a vector that I was given. Sure, an arrow can begin at the origin and extend outward, but not necessarily. Any arrow representing a vector is the same as any other as long as it has the same length and direction, no matter where the base of the arrow sits, the zero vector should still be the zero vector.
So I ask… How can the zero vector not be in any plane, if it is indeed properly understood as "pick a point, move 0 units on the x-axis, 0 units on the y-axis, and 0 units on the z-axis"?
Best Answer
From your question it appears you are confused with the meaning of the word vector. You describe a vector in, say, $\mathbb R^2$ is a displacement operation with an arbitrary starting point. However, a it's not quite that. We consider two such displacements to be essentially the same if they have the same direction and the same magnitude. This essential sameness defines an equivalence relation on the set of all such displacement. In that context a vector is an equivalence class, not just a representative of it.
The modern approach is to abandon these inconveniences all together and adopt an axiomatic approach. A vector space is a set with extra structure satisfying a certain list of axioms. Then, a vector is, by definition, an element of a vector space. The zero vector is then the (provably) unique vector in a given vector space which behaves neutrally with respect to addition of vectors. That is what the zero vector is. It should be emphasized that the precise name of the zero vector is highly sensitive to the vector space structure. For instance, $\mathbb R$ with its usual vector space structure admits $0$ as the zero vector. However, for every $a\in \mathbb R$ it is possible to endow $\mathbb R$ with a vector space structure such that $a$ is the zero vector. This is what happens when we shift from a definition of what vectors are (upon which your understanding of vectors currently relies) to not caring about what they are and only caring about what you can do with them (this is the axiomatic approach). There is good reason why the latter is the prevalent choice in modern mathematics. Nobody cares, nor should we care about what something is. All that matters is what we can do with it. It saves a lot of headaches and endless philosophical quarrels if you don't even attempt to define what something is and instead simply resort to listing the things you can do with these things.
With all that said, other than the cases you mention where the zero vector does not belong to a given set, consider the empty subset of any vector space. It is never a vector subspace since it does not contain the zero vector (nor any other vector).