Why $K^0 = \{0\}$?

Question

I am reading "Linear Algebra" by Takeshi SAITO.

Why $n \geq 0$ instead of $n \geq 1$?
Why $K^0 = \{0\}$?
Is $K^0 = \{0\}$ a definition or not?

He wrote as follows in his book:

Let $K$ be a field, and $n \geq 0$ be a natural number.

$$K^n = \left\{\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} \middle| a_1, \cdots, a_n \in K \right\}$$

is a $K$ vector space with addition of vectors and scalar multiplication.

$$\begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} + \begin{pmatrix} b_{1} \\ b_{2} \\ \vdots \\ b_{n} \end{pmatrix} = \begin{pmatrix} a_{1}+b_{1} \\ a_{2}+b_{2} \\ \vdots \\ a_{n}+b_{n} \end{pmatrix}\text{,}$$ $$c \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} = \begin{pmatrix} c a_{1} \\ c a_{2} \\ \vdots \\ c a_{n} \end{pmatrix}\text{.}$$ When $n = 0$, $K^0 = 0 = \{0\}$.

Answer 1

It is a convention, which you can take as a definition.

Since $K^n$ is an $n$-dimensional vector space when $n$ is a positive integer, we would like to have $K^0$ to denote a zero-dimensional vector space, which would be $\{0\}$.

Answer 2

It can be thought of as a "useful" definition. Any subspace of $K^n$ is isomorphic to $K^m$ for some $m\leq n$. If you don't define $K^0=\{0\},$ then this isn't true for the $0$-subspace.

Another approach is to define $K^n$ as the set of functions from a set of $n$ elements to $K$. When $n=0$, the set of functions from the empty set to any set is $1.$

It's worth noting that the three occurences of $0$ in the equality $K^0=0=\{0\}$ are representing three different things.

The first zero is the natural number $0.$

The second is a trivial space, a vector space with one element.

The third $0$ is the element of that trivial space.

You might then write it as:

$$K^0=\mathbf 0=\{\vec 0\}$$

Answer 3

Another way to see as why this definition is natural is that $K^n$ can be seen as the set of functions from a set with $n$ elements to $K$ (and the obvious "pointwise" addition, multiplication by scalar etc).

This way, $K^0$ must be the set of functions from a set of zero elements (empty set) to $K$. This is a set with only one element, which must be the zero element.