Why $K^0 = \{0\}$? 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 $n = 0$, $K^0 = 0 = \{0\}$.
 A: 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 $\{0\}$.
A: 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:
$$K^0=\mathbf 0=\{\vec 0\}$$
A: 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.
