Linear Algebra: Vector Basis (change of)

[p1ayaone1]

Homework Statement

Given one known orthonormal basis S in terms of the standard basis U, how would I express a third basis T in terms of U when I know its representation in S?

For example, U consists of
<1,0,0>
<0,1,0>
<0,0,1>

And S consists of (for example)
<0.36, 0.48, -0.8>
<-0.8, 0.6, 0>
<0.48, 0.64, 0.6>

(Note all colums and rows have unit magnitude and are orthogonal)

The values of S are in x, y, and z components of U.

Now I want T in terms of U, when I know that in terms of S it is
<1, 0, 0>
<0, -1, 0>
<0, 0, -1>

I'd like to arrive at a general solution, i.e. one that works for any T (so long as it's orthonormal).

Homework Equations

(sorry I don't know Latex)

Point (in frame A) = (BtoA)RotationMatrix * Point (in frame B)

The Attempt at a Solution

These bases I'm talking about are components of rigid-body frames, so each of them has an "origin", which is transformed through elementwise linear combination. (i.e. the origin of S is Sx, Sy, Sz, so add (0, 0, 0) to (Sx, Sy, Sz). Similarly, T has origin (Tx, Ty, Tz), which means that its origin in terms of U is (0+Sx+Tx, 0+Sy+Ty, 0+Sz+Tz).

That is fine and good for purely translated frames, but I need now to consider rotations. T and S share the same X-direction, but the Z (hence Y to maintain right-handedness) directions are inverted.

I know I can define a rotation matrix to map a single point from one frame to another (I even know how to use the 4x4 general transformation matrix to get translation&rotation in one shot) but that's not quite the problem I'm facing (I think).

What I would like to do is express a direction&orientation that I know in one orthonormal basis (T known in S) in another (T unknown in U but S known in U).

btw, this isn't a homework problem but I feel this forum is the most likely to generate a solution. So it's possible the problem statement needs work.

Thanks for your input!

 

[p1ayaone1]

Here's me answering my own question...

The components of a vector V in the standard basis U can be interpreted as the projections of that vector onto the basis unit vectors. All I need to do is project the same vector onto those vectors representing S (in terms of U), and extract the corresponding components.

I brought the problem into R^2 to solve, so for example I have a vector V = <-2,1> in terms of U. Projecting <-2, 1> onto <1, 0> and <0, 1>, I get <-2, 0> and <0, 1>, respectively.

Now if S is in terms of U, (i.e. the x-prime axis is some arbitrary unit vector in U, and y-prime is right-orthogonal to it) all I need to do is project V = <-2, 1> onto x-prime and y-prime. Now I have the same vector described in two coordinate systems.

The best thing is that I'm only using the projection "operator", which is easily extended to R^3. No need to "change the basis".