Serlo: EN: Coordinate spaces

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The coordinate space is the vector space of $n$-tuples $(x_1,...,x_n)$ with entries $x_k\in K$ in a field $K$, equipped with componentwise addition and scalar multiplication. An example is the vector space ${ℝ}^3$ known from school, with vectors $(x_1,x_2,x_3)$ and $K=ℝ$.

Derivation

In mathematics, one often uses existing structures to define new and more general ones. As we have already seen in introduction to vector space, we can extend from the vector spaces ${ℝ}^3$ and ${ℝ}^2$ over the real numbers $ℝ$ to more general vector spaces ${ℝ}^n$ for every natural number $n$. For this we recall how the addition of two vectors and the scalar multiplication between a vector and a scalar works in ${ℝ}^3$ and ${ℝ}^2$: We have

In other words, the addition and scalar multiplication are defined component-wise. That is, we perform the addition and the scalar multiplication in ${ℝ}^2$ and ${ℝ}^3$ by adding in each component and multiplying in each component with the scalar, respectively. In the same way we can define an addition and a scalar multiplication if our vectors do not consist of two or three but of $n$ real numbers. That is, on the set

we define a component-wise vector addition and a scalar multiplication:

For the vector addition we use the real number addition and for the scalar multiplication the real number multiplication. Let $x=(x_1,\dots ,x_n)$ and $y=(y_1,\dots ,y_n)$ with $x_i,y_i\in ℝ$, then the vector addition is defined by

Let $x\in {ℝ}^n$ and $\alpha \in ℝ$, then the scalar multiplication is defined by

We can now easily verify that ${ℝ}^n$ with this vector addition and scalar multiplication is a vector space over the field $ℝ$.

Thus we have transferred the known structure of the real numbers $ℝ$ and the vector spaces ${ℝ}^2$ and ${ℝ}^3$ to the vector space ${ℝ}^n$. We also refer to ${ℝ}^n$ as coordinate space of dimension $n$ over $ℝ$.

Simple generalization

If we look again at the definition of the vector space structure on ${ℝ}^n$, we have used only multiplication and addition on $ℝ$. But now, any field $K$ admits a multiplication and addition. Thus the above construction also provides us with a way to define a coordinate space over arbitrary fields. This coordinate space is defined by taking the set

and equipping it with an addition and a scalar multiplication. For this we copy the definition of above and define it component-wise. That means we use in every component the addition and multiplication of $K$ to define the addition and scalar multiplication on $K^n$.

Definition: coordinate space

Now, we equip the set $K^n$ with an addition and a scalar multiplication.

Mathe für Nicht-Freaks: Vorlage:Definition

Coordinate spaces are vector spacesVorlage:Anker

In the article introduction to the vector space we used the above construction first over $ℝ$ and then over arbitrary fields to derive the vector space axioms. Moreover, field satisfies similar properties as vector space and we have used the former very directly to define addition and scalar multiplication on the coordinate space. Therefore, we can conjecture that the definition of $⊞$ and $⊡$ on $K^n$ defines a vector space structure, as well. And this is indeed true, as we will verify now.

Thus we have shown all eight vector space axioms and hence $(K^n,⊞,⊡)$ is indeed a $K$-vector space. }}

Relation to the field being a vector space

We have already seen that $K$ is a $K$-vector space. This is a special case of the coordinate spaces $K^n$, because it is $K^1=K$. Here we take the vectors $(x)\in K^1$ to be elements of the field. We then write instead of the $1$-tuple $(x)$ only $x$, instead of $(1)$ only $1$ and instead of $(0)$ only $0$.

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