If the set B containing the vectors,
b sub 1 through b sub n
is an ordered basis for a vector space V
and vector X is a vector in V,
then vector X is equal to some linear combination
of the basis vectors b sub 1 through b sub n.
The scalers C sub 1 through C sub n
are called the coordinates of vector X
relative to the basis B.
The coordinate matrix or a coordinate vector
of vector X relative to B can be denoted two common ways.
First, we have the coordinates of vector X
relative to the basis B
or we have the coordinates of relative to base B of vector X
equals the column matrix containing the scalers C sub 1
through C sub n.
So let's take a look at an example.
Let's assume the gray triangle is a ramp.
First we have the coordinate system
relative to the standard basis vectors
which are the vectors 1,0 and 0,1.
If we wanted to sum two vectors to reach the black point
where one vector is along the ramp
and the other is orthogonal to the ramp,
we can use these two vectors
where the vector along the ramp has coordinate 8,4
relative to the standard basis vectors.
And the second vector is the vector -2,4
again, relative to the standard basis vectors.
And the sum of these two vectors
would be the vector from the origin to the black point,
this point here.
Some of the two vectors notice how the coordinates are 6,8.
This vector has coordinates 6,8
relative to the standard basis vectors.
However, at times it can be helpful
to have a coordinate system
that will be parallel and perpendicular to the ramp
shown here on the right.
And we can create this coordinate system
by using the vectors 8,4 and -2,4
as basis vectors.
Using these basis vectors for the coordinate system,
the red vector along the ramp now has coordinates 1,0
relative to base B because this vector is one times
the first basis vector of 8,4.
And the second vector that is orthogonal to the ramp
now has coordinates at 0,1
because this vector is one times
the second basis vector of -2,4.
Using this coordinate system,
the vector from the origin to the black point
would be the sum of these two vectors
which would give the coordinates 1,1.
These are the coordinates for this vector
relative to base B.
And now let's relate the two coordinate systems.
If we wanted to go from the vector 1,1
relative to base B to the vector 6,8
relative to the standard basis vectors,
we would multiply 1 times the first basis vector of 8,4
plus 1 times the second basis vector of -2,4
which gives us the coordinates 6,8
relative to the standard basis vectors
which is what we expect.
We also want to point out how we can write
the sum of these products as the two by two matrix formed
using the basis vectors times the coordinates of the vector
relative to base B,
which does result in the vector 6,8
which can have the coordinates relative
to the standard basis vectors.
We call this two by two matrix
the transition matrix from B to S
and we use the variable P for the matrix.
And now if we wanted to go from the coordinates 6,8
relative to the standard basis vectors
to a vector using the coordinates relative to base B,
we would need the transition matrix from S to B
which is equal to the inverse
of the transition matrix from B to S.
So the coordinates of vector X relative to base B
is equal to, again, the inverse
of the transition matrix from B to S
which we have here on the left
times the coordinates of the vector
relative to the standard basis vectors,
which for this example gives us the coordinates 1,1
which are the coordinates for the vector relative to base B.
So let's go back to our notes and summarize.
To convert a vector from coordinate relative to base B,
to coordinate relative to the standard basis vectors,
we need to find the product of the transition matrix
from B to S which is formed using the basis vectors
from set B and the coordinates of vector X
relative to base B.
Again, this matrix P from B to S
is called the transition matrix from B to S.
And a converted vector X
from coordinates relative to the standard basis S
to the coordinate relative to basis B,
we solve the equation above
for the coordinates of vector X relative to base B
which involves finding the inverse
of the transition matrix from B to S
which we see here on the right.
The coordinates of vector X relative to B
is equal to the inverse of the transition matrix
from B to S times the coordinates of vector X
relative to the standard basis S.
So this inverse matrix
is equal to the transition matrix from S to B
which is what we need to go
from coordinates in S to coordinates in B.
Let's take a look at two examples.
If the coordinates of vector X relative to base B are -2,5
then what are the coordinates of vector X
relative to the standard basis vectors?
We can do this two ways.
Using the definition we know the coordinates of vector X
relative to the standard basis vectors
is equal to the first coordinate of -2
times the first basis vector of 1,1
and then plus the second coordinate of 5
times the second basis vector of 1,-1.
Simplifying, we have -2 times 1 plus 5 times 1 which is 3.
And then we have -2 times 1 plus 5 times -1 which is -7.
We can also find the coordinates of vector X
relative to the standard basis vectors
by determining the transition matrix from B to S
and then multiplying by the coordinates of vector X
relative to base B.
The transition matrix is formed using the basis vectors
from B and therefore the first column is 1,1
the second column is 1,-1
and we are given the coordinates of vector X
relative to base B which are -2,5.
And this product will give us the same result
of the vector 3,-7
which are the coordinates of the vector
relative to the standard basis vectors.
And then number two, we want to go in the opposite direction.
We are given the coordinates of vector X
using the standard basis vectors are 4,-8
and we're trying to find the coordinates of vector X
relative to base B.
So the coordinates of vector X
relative to base B is equal to
the transition matrix from S to B
times the coordinates of vector X
relative to the standard basis vectors
and the transition to matrix from S to B
is equal to the inverse of the transition matrix from B to S
times the coordinates of vector X
relative to the standard basis vectors.
So we have the inverse of the matrix above
which is the matrix 1, 1, 1, -1
and then times the coordinates of the vector
relative to the standard basis vectors
which we know are 4,-8.
So the next step is to find the inverse of this matrix,
which I've already done to save a little bit of time.
The first row was 1/2, 1/2
and the second row was 1/2, -1/2.
Multiplying, we have 1/2 times 4 plus 1/2 times -8.
That's 2 plus -4 which is -2.
And then for the second entry,
we have 1/2 times 4 plus -1/2 times -8
which is 2 plus 4, which is 6.
I hope you found this helpful.