Transformation matrix problem

[muckymotter]

Consider the quadrilateral (namely Q) in R^3 formed by the points

(1, 0, 0), (2, 0, 0), (1, 1, 3), and (2, 1, 3).

 

(a) What should the coordinates be for the figure R we get by rotating Q counterclockwise in the x-y plane by 45 degrees, then dilating it by a factor of 3/2, then translating it along the vector (-2, 1, -1)?

 

 

Okay, so what I did was I used the matrix

 

cos45 -sin45 0

sin45 cos45 0

0 0 1

 

 

After this, multiplied the identity matrix for R^3 by 3/2 and then multiplied it by the matrix with 45 substituted for theta.

 

Then T(x,y,z)=(-2+3sqrt(2)x/4-3sqrt(2)y/4,1+3sqrt(2)x/4+3sqrt(2)y/4,-1+3z/2)

 

I substituted each of the quadrilateral points in for T(x, y, z) to come up with the four points and got:

 

 

(3/(2sqrt2) - 2, 3/(2sqrt2) + 1, -1),

(3/sqrt2 - 2, 3/sqrt2 + 1, -1),

(-2, 3/sqrt2 + 1, 3.5),

(3/(2sqrt2) - 2, 9/(2sqrt2) + 1, 3.5)

 

But my trouble lies in doing (b) so if anyone could explain how it is done, I would be really grateful as I have been stuck on this for days!:

(b) What matrix transforms Q into R?

[Bignose]

Assuming that that answer for a is right (I didn't check, and I assume you have checked), you've almost done the entire answer. Let me call the first matrix Q and second matrix (your answer) R. These are related by Q = H*R where here H is the transformation matrix. Well, to isolate H, you just right multiply both sides of the this equation by the inverse of R, R^-1.

 

Since a matrix times its inverse is the identity matrix on the right hand side of the equation you get H times the identity matrix. Then, any matrix times the identity matrix is itself, so the right hand side is just H.

 

The equation becomes Q*R^-1 = H.

 

So, you just need to get R^-1 and multiply Q by R^-1 to get H, the transformation matrix you seek.

[muckymotter]

Thanks! I'm a little confused, though -

 

The first matrix Q is a 3x3 matrix, but the transformation matrix R is a 4x3 so I can't multiply them.

[Bignose]

Well, there is a generalized matrix inverse also known as pseudoinverse for non-square matrices. The pseduoinversed matrix R would be a 3x4 matrix, and you can right mulitply a 3x3 matrix by a 3x4 matrix.