Let be a vector space over the field
having dimension .
Let be a basis for .
We call an ordered basis for .
By fixing an order on the basis elements, we can now represent
every vector in as an -tuple.
More precisely, let . Recall that there exist scalars
such that
Define to be the -tuple
.
We call a tuple representation
(or coordinate representation of
with respect to the ordered basis .
Clearly, if we choose a different basis or a different ordering of the basis
vectors, we get a different representation for the same vector. That is
why it is so important to specify the ordered basis when working with
tuple representations.
Examples
Note that is an
ordered basis for .
Let .
What is ?
Solution.
We first write as a linear combination of the ordered basis elements.
That is, we want to find real numbers and such that
.
Thus, we need to solve the system
Solving gives and .
Hence, .
Let denote the vector space of polynomials in with
real coefficients having degree at most .
Let denote the ordered basis
and let denote the ordered basis
. (Check that these are indeed ordered bases.)
We want to find the tuple representations of
with respect to these ordered bases.
To determine ,
we need to find real numbers
such that
Simplifying the right-hand side, we obtain
Since the left-hand side has no term, we must have
. Now, comparing the coefficients of on both
sides, we get . Since the left-hand side has
no constant term, we must have .
In other words, , giving
.
For ,
note that . Hence,
.
Working with tuple representations
The beauty of these tuple representations of vectors is that we can now
work with tuples instead. For instance,
if and is an ordered basis for , then
one can easily check that
In other words, adding two vectors in simply requires adding
the corresponding tuple representations
in .
And if is a scalar,
then . Thus,
multiplying a vector by a scalar simply involves multiplying the corresponding
tuple representation by the same scalar.