Let the vectors in sets B and C
be two bases for R two.
We're asked to find the change of coordinates matrix
or transition matrix from B to C.
The transition matrix or change of coordinates matrix
from B to C
is equal to the product of the transition matrix
from S to C.
And the transition matrix from B to S.
The right side is a,
the right side is a composition of transformations.
And therefore we'd work our way from right to left.
The product results in the transition from B to S
and then S to C, resulting in the transition matrix
from B to C.
And the transition matrix from S to C
is equal to the inverse of the transition matrix
from C to S.
And we still have times, the transition matrix from B to S.
Well, the inverse of the tradition matrix from C to S
is equal to the inverse of the matrix formed
using the basis vectors in set C.
And the transition matrix from B to S
is equal to the matrix formed
using the basis vectors in set B.
Which means this is equal to the inverse
of the two by two matrix formed
using the basis vectors from set C.
And therefore the first column is one zero.
Second column is negative three one,
and then we have times the transition matrix from B to S,
which is the two by two matrix formed
using the basis vectors in set B.
And therefore the first column is two, three.
And the second column is one, two.
The next step is to find the inverse
of the two by two matrix,
which I've already done to save time.
And it's a pretty simple inverse matrix to find.
The first row is one, three,
and the second row is zero one.
And then we still have times the transition matrix
from B to S.
Multiplying the entry in row one, column one
is one times two, plus three times three, which is 11.
The entry in row one column two is one times one,
plus three times two, which is seven.
The entry in row two, column one is zero times two,
plus one times three, which is three.
And the entry in row two, column two is zero times one,
plus one times two, which is two.
This is the transition matrix,
or change of coordinates matrix, from B to C.
There's a second part to this question.
If the coordinates of vector X relative to basis B
are two negative three,
we're asked to find the coordinates of vector X
relative to basis C.
So the coordinates of vector X
relative to basis C is equal to the transition matrix
from B to C
times the coordinates of vector X
relative to basis B.
And therefore the coordinates of vector X
relative to basis C is equal to the product
of the transition matrix where the first row is 11, seven.
And the second row is three, two times
the coordinates the vector X relative to basis B,
which are given as two, negative three.
So the first coordinate is 11 times two,
plus seven times negative three, which is 22,
plus negative 21, which is one.
And the second coordinate is three times two,
plus two times negative three, which is zero.
The coordinates of vector X
relative to basis C are one, zero.
I hope you found this helpful.