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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
.
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
.
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 (The set
)
Let
and
be a field.
We define
We denote the elements of this set as
-tuples with entries in
.
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)
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.
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.
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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
.
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