5

Reading through some lecture notes, I found out the first example of a vector space is the zero vector space, which contains only the zero vector.

I understand that it meets all the axioms of a vector space, but why do we need it? Why bother mentioning that it is a vector space? Particularly, in what way is it useful while answering questions about Linear Algebra?

AnalysisShmalysis
  • 127
  • 1
    Welcome to Mathematics Stack Exchange. You might likewise wonder why we need an empty set – J. W. Tanner Mar 10 '20 at 02:31
  • 3
    Id offer up three reasons. 1) That we talk about it for the sake of completion. 2) As proof of existence by example. 3) As the base case for inductive arguments. There may be other reasons. – CogitoErgoCogitoSum Mar 10 '20 at 02:34
  • 10
    Kernels are important and so are injective maps. – Randall Mar 10 '20 at 02:50
  • 1
    An important concept in linear maps (and in abstract algebra) is the kernel. The set of vectors that map onto the zero vector. Without a valid definition of the zero vector it is difficult to discuss this space. – user317176 Mar 10 '20 at 02:51
  • @DougM OP asks about the usefulness of the zero subspace, not of the zero vector. You only need the zero vector to define kernels, technically. – Federico Poloni Mar 10 '20 at 10:58

2 Answers2

8

Take a vector space $V$ of many dimensions. The intersection of two subspaces of $V$, let's call them $W_1$ and $W_2,$ is also a subspace.

Sometimes the intersection of two subspaces $W_1$ and $W_2$ contains only the zero vector. If the set containing only the zero vector were not considered a vector space, then what I said in my first paragraph would be false.

So one reason why the vector space containing only the zero vector is useful while answering questions about Linear Algebra is that it saves us from having special cases that we have to give special treatment to. We would rather write "the intersection of $W_1$ and $W_2$ is a subspace" than write "the intersection of $W_1$ and $W_2$ is either a subspace or a set containing only the zero vector."

David K
  • 98,388
  • 2
    Great answer! To add to it, there are many other settings where one would have to write "a subspace or {0}" if {0} were not considered a subspace: defining kernels and images, for instance. – Federico Poloni Mar 10 '20 at 10:56
  • The same argument works for many mathematical structures. Often, as a result of some operation (e.g. intersection of two structures, kernel of a structure-preserving function aka homomorphism, ...) we would like to get again a structure of the same kind. Thus, subsuming special cases like the empty structure serves ease in building up the theory. – ComFreek Mar 10 '20 at 11:43
5

Here is a frequently-occurring use of the $\{0\}$ subspace: we often wish to speak of a vector space $V$ as the direct sum of two subspaces $X$ and $Y$, written as

$V = X \oplus Y; \tag 1$

this means that every

$v \in V \tag 2$

may be expressed in the form

$v = x + y, \; x \in X, y \in Y; \tag 3$

condition (3) is written

$V = X + Y; \tag 4$

(1) requires the additional hypothesis

$X \cap Y = \{0\}; \tag 5$

this ensures the decomposition (4) is unique: if

$x_1 + y_1 = x_2 + y_2, \tag 6$

then

$X \ni x_1 - x_2 = y_2 - y_1 \in Y; \tag {6.6}$

thus if (5) binds, we may affirm that

$x_1 - x_2 = 0 = y_2 - y_1, \tag 7$

or

$x_1 = x_2, y_1 = y_2. \tag 8$

The construction (1) is so useful, and arises so frequently, that the introduction of $\{0\}$ is justified by this alone. Furthermore, $\{0\}$ satisfies all the vector space axioms, so not admitting it creates yet one more exception, which if nothing else creates more to remember.

Robert Lewis
  • 71,180