In a vector space, any vector can be written as a linear combination of a basis. The coefficients of the linear combination are called the coordinates of the vector with respect to the basis.
Previously, we have provided two definitions of a vector space:
informal definition: a vector is a finite array of numbers, and a set of such arrays is said to be a vector space if and only if it is closed with respect to taking linear combinations;
formal definition: a vector space is a set equipped with two operations, called vector addition and scalar multiplication, that satisfy a number of axioms.
The simpler informal definition is perfectly compatible with the more formal definition, as a set of numerical arrays satisfies all the properties of a vector space, provided that vector addition and scalar multiplication are defined in the usual manner and that the set is closed with respect to linear combinations.
We now introduce a new concept, that of a coordinate vector, which makes the two definitions almost equivalent: if we are dealing with an abstract vector space, but its dimension is finite and we are able to identify a basis for the space, then we can write each vector as a linear combination of the basis; as a consequence, we can represent the vector as an array, called a coordinate vector, that contains the coefficients of the linear combination.
Once we have obtained this simple representation, we can apply the usual rules of matrix algebra to the coordinate vectors, even if we are dealing with an abstract vector space.
Not only this is very convenient, but it blurs the differences between the two approaches to defining vectors and vector spaces (at least for the finite-dimensional case).
We are now ready to give a definition of a coordinate vector.
Definition
Let
be a finite-dimensional linear space. Let
be a basis for
.
For any
,
take the unique set of
scalars
such
thatThen,
the
vectoris
called the coordinate vector of
with respect to the basis
.
Note that the uniqueness of the scalars
is guaranteed by the
uniqueness of
representations in terms of a basis.
Let us make some examples.
Let
be a vector space and
a basis for it.
Suppose that a vector
can be written as a linear combination of the basis as
follows:
Then, the coordinate vector of
with respect to
is
Consider the space
of second-order
polynomialswhere
the coefficients
and the argument
are scalars.
As we have already discussed in the lecture on
linear spaces,
is a vector space provided that the addition of polynomials and their
multiplication by scalars is performed in the usual manner.
Consider the
polynomials
These three polynomials form a basis
for
because they are linearly
independent (no combination of them is equal to zero for any
)
and they can be linearly combined so as to obtain any
of the form
above:![[eq12]](/images/coordinate-vector__30.png){kind=small}
The coordinate vector of
with respect to the basis we have just found
is
The addition of two vectors can be carried out by performing the usual operation of vector addition on their respective coordinate vectors.
Proposition
Let
be a linear space and
a basis for
.
Let
.
Then, the coordinate vector of
with respect to the basis is equal to the sum of the coordinate vectors of
and
with respect to the same basis, that
is,
Suppose that the representations in terms of
the basis
areso
that the coordinate vectors
are![[eq17]](/images/coordinate-vector__42.png)
By
the commutative and distributive properties of vector addition and scalar
multiplication in abstract vector spaces, we have
that![[eq18]](/images/coordinate-vector__43.png)
Thus,
the coordinate vector of
is
The multiplication of a vector by a scalar can be carried out by performing the usual operation of multiplication by a scalar on its coordinate vector.
Proposition
Let
be a linear space and
a basis for
.
Let
and let
be a scalar. Then, the coordinate vector of
with respect to the basis is equal to the product of
and the coordinate vector of
,
that
is,
Suppose the representation in terms of the
basis
isso
that the coordinate vector
isBy
the associative and distributive properties of scalar multiplication in
abstract vector spaces, we have
thatThus,
the coordinate vector of
is
When the elements of a linear space
are one-dimensional arrays of numbers (vectors, in the simplest sense of the
term), then they coincide with their coordinate vectors with respect to the
standard basis.
Example
Let
be the space of all
column vectors. Let
be its canonical basis, where
is a vector whose entries are all
,
except the
-th,
which is equal to
:
Take
any
Then,
is the same as its coordinate vector with respect to the basis
,
that
is,because
Below you can find some exercises with explained solutions.
Let
be the vector space of all third-order polynomials.
Perform the addition of two
polynomialsandby
using their coordinate vectors with respect to the
basis
Check that the result is the same that you would get by summing the two polynomials directly.
The representations in terms of the basis
are![[eq36]](/images/coordinate-vector__79.png)
Thus,
the two coordinate vectors
are![[eq37]](/images/coordinate-vector__80.png)
Their
sum
is![[eq38]](/images/coordinate-vector__81.png)
so
that![[eq39]](/images/coordinate-vector__82.png)
This
is the same result that we obtain by carrying out the addition
directly:![[eq40]](/images/coordinate-vector__83.png)
Let
be the space of all
vectors.
Consider the basis
where
Find the coordinate vector of
with
respect to the given basis.
We have
thatTherefore,
the coordinate vector of
is
Please cite as:
Taboga, Marco (2021). "Coordinate vector", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/coordinate-vector.
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