Linear Algebra: Vector Basis (change of)

In summary, To express a third basis T in terms of the standard basis U, when given its representation in another orthonormal basis S, one can project the basis unit vectors of T onto those of S in terms of U to obtain the corresponding components of T in terms of U. This method can be extended to higher dimensions and avoids the need for changing the basis.
  • #1
p1ayaone1
74
0

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!
 
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  • #2
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".
 

1. What is a vector basis in linear algebra?

A vector basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the space can be expressed as a linear combination of the basis vectors, with unique coefficients.

2. Why is a vector basis important in linear algebra?

A vector basis is important in linear algebra because it allows us to represent and manipulate vectors in a more efficient and organized manner. It also serves as a foundation for many important concepts and operations in linear algebra, such as matrix multiplication and change of basis.

3. How is a vector basis related to change of basis in linear algebra?

A vector basis is essential for understanding change of basis in linear algebra. When we change the basis of a vector space, we are essentially expressing the same vectors in terms of a different set of basis vectors. This process involves finding the coefficients that relate the original basis to the new basis, which can be represented by a matrix known as the change of basis matrix.

4. Can a vector basis be non-unique?

Yes, a vector basis can be non-unique. This means that there can be more than one set of linearly independent vectors that can span a vector space. However, any two basis sets for the same vector space will have the same number of elements, known as the dimension of the space.

5. How do you determine if a set of vectors form a basis for a vector space?

To determine if a set of vectors form a basis for a vector space, we need to check two conditions: linear independence and spanning. Linear independence means that none of the vectors in the set can be expressed as a linear combination of the other vectors. Spanning means that the vectors can be used to express any vector in the space. If both conditions are met, then the set of vectors is a basis for the vector space.

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