Coordinate spaces – Serlo

Aus Wikibooks
Zur Navigation springen Zur Suche springen

The coordinate space is the vector space of -tuples with entries in a field , equipped with componentwise addition and scalar multiplication. An example is the vector space known from school, with vectors and .

Derivation[Bearbeiten]

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 and over the real numbers to more general vector spaces for every natural number . For this we recall how the addition of two vectors and the scalar multiplication between a vector and a scalar works in and : 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 and 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 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 and with , then the vector addition is defined by

Let and , then the scalar multiplication is defined by

We can now easily verify that with this vector addition and scalar multiplication is a vector space over the field .

Thus we have transferred the known structure of the real numbers and the vector spaces and to the vector space . We also refer to as coordinate space of dimension over .

Simple generalization[Bearbeiten]

If we look again at the definition of the vector space structure on , we have used only multiplication and addition on . But now, any field 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 to define the addition and scalar multiplication on .

Definition: coordinate space[Bearbeiten]

Definition (The set )

Let and be a field. We define

We denote the elements of this set as -tuples with entries in .

Example (Examples for tuples)

As an example, is a -tuple with entries in (we also just say tuple or pair instead of -tuple).

is a -tuple with entries in (we also say triple instead of -tuple).

Now, we equip the set with an addition and a scalar multiplication.


Definition (vector space operations on )

The addition is defined by

Similarly, we define the scalar multiplication by

We call coordinate space.

Hint

We have so far taken a tuple to be a row vector. That means, we have written for an element in . Just as well, instead of writing the elements as one row with columns, we could also write them as one column with entries. Then an element in would look like this:

This other representation does NOT change the properties of as vector space. If we take the vector to be a row with columns, then is called a row vector. If we take in turn as a column with rows as a column vector.

It will prove useful for matrices (missing) to write the vectors in as column vectors. Therefore, from now on, we will work with column vectors. However, the notation of a column vector in a line is not very space-saving. Therefore, we introduce the following notation: Instead of

we write the vector as . The symbol means that this vector is transposed, i.e. the row vector is transformed into a column vector. This transposition is the same as for matrices (missing)

Coordinate spaces are vector spaces[Bearbeiten]

In the article introduction to the vector space we used the above construction first over 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 defines a vector space structure, as well. And this is indeed true, as we will verify now.

Theorem ( is a vector space)

is a -vector space.

How to get to the proof? ( is a vector space)

We proceed as in the article "Proofs for vector spaces", where the eight vector space axioms were checked, one by the other.

The definitions of are chosen by copying the operations in the field "in a natural (component-wise) way to the vector space ." We show that the vector space axioms follow directly from the corresponding field axioms. So we have to show in every step that the definitions of and induce the known properties of and in the field .

Proof step: Associativity of addition

Let . Then, we have:

This shows the associativity of addition.

Proof step: Commutativity of addition

Let . Then, we have:

This shows the commutativity of addition.

Proof step: Neutral element of addition

We still need to show that there is a neutral element for which we have that

Since we trace all properties back to the corresponding properties in , here we use the neutral element of addition to construct the neutral element of addition . That means, we set

Is this neutral element of addition really neutral? For this, we have to check : Let . Then, we have:

Thus we have shown that is the neutral element of addition.

Proof step: Inverse with respect to addition

Let .

We need to show that there exists a such that . Let us reduce this problem to the properties of arithmetic operations in . In we have that if and , then . Therefore, for we choose the tuple as the potential inverse. Then, we have:

Thus we have shown that for any there exists a with .

Proof step: Scalar distributive law

Let and . Then, we have:

Thus the scalar distributive law is also shown.

Proof step: Vectorial distributive law

Let and . Then, we have:

This establishes the vectorial distributive law.

Proof step: Associativity of multiplication

Let and . Then, we have:

This shows the associative law for multiplication.

Proof step: unitarity law

Let . Then, we have:

Thus we have also shown the unitary law.

Thus we have shown all eight vector space axioms and hence is indeed a -vector space. }}

Relation to the field being a vector space[Bearbeiten]

We have already seen that is a -vector space. This is a special case of the coordinate spaces , because it is . Here we take the vectors to be elements of the field. We then write instead of the -tuple only , instead of only and instead of only .