Why do we need a zero vector space?

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."

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.