guys. I'm not entirely sure how I'm not getting the right answer for this question. I'll try to explain what I've tried so far.
I need to computer MB1->B2 and MB2->B1
B1 = {(0,0,1),(1,0,0),(0,1,0)}
B2 = {(1,0,1),(1,1,0),(0,1,1)}
For MB1->B2,
I used B2 inverse multiplied by B1 both in matrix form.
For MB2 -> B1
I used B1 inverse multiplied by B2 both in matrix form.
But then I need to verify that:
M1 = MB2->B1 * M2 * MB1->B2
M2 = MB1->B2 *M1 * MB1->B2
where M1 and M2 are associated to phi star with respect to B1 and B2.
For M1 and M2, I used the inverse of B1 and B2 in matrix form, but I'm not getting the right answer. I'm receiving M1 = MB2->B1 * M1 * MB1->B2.
Thank you! I'm sorry that I couldn't write it out in matrix form. I tried using the guide, but I couldn't make much sense of it.
Best Answer
I assume you want to compute the change-of-basis-matrix from basis B1 to basis B2.
The first step is to write the new basis vectors as a linear combination of the old basis vectors:
$$(1,0,1)=1\cdot(0,0,1)+1\cdot(1,0,0)+0\cdot(0,1,0)$$
$$(1,1,0)=0\cdot(0,0,1)+1\cdot(1,0,0)+1\cdot(0,1,0)$$
$$(0,1,1)=1\cdot(0,0,1)+0\cdot(1,0,0)+1\cdot(0,1,0)$$
The first column in the change-of-basis-matrix is the coordinate of the first new basis vector with respect to the old basis vectors. Repeat this for the second and third column. The you get the following change-of-basis matrix T: \begin{equation} \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{bmatrix} \end{equation}
The general formula to transform new coordinates into old coordinates and vice versa:
coordinate of a point with respect to old basis = $T$ multiplied by coordinate in new basis
coordinate of a point with respect to new basis = $T^{−1}$ multiplied by coordinate in old basis
Remark: T translates from the new basis to the old basis!