Linear Algebra- Transformations and

[KyleS4562]

1.
Question:
Which of the following linear transformations T from |R^3 to |R^3 are invertible? Find The inverse if it exists.

a. Reflection about a plane
b. Orthogonal projection onto a plane
c. Scaling by a factor of 5
d. Rotation about an axis

Homework Equations

The Attempt at a Solution

Well I know only a, c, and d are invertible but I do not know how to go about finding their inverses. Only i have assumed so far it involves some vector <x,y,z> about some arbitrary thing such as in part a, a plane through the origin with an arbitrary normal <a,b,c> but I am not even sure about that.

 

[Dick]

Hi KyleS4562, welcome the Forums. I think they want a fairly abstract answer to each question, since they didn't describe most of them in enough detail to let you give a formula like <x,y,z> transforms to whatever. Just answer in the same spirit they asked. What's the inverse of a reflection? What's inverse of an expansion by 5? What's the inverse of a rotation? Just answer in words.

 

[KyleS4562]

Thanks... i have about a page and a have worth of random row reducing trying to come up with a generic inverse matrix for a with any plane with any given <a,b,c> normal and I still not got the final matrix so I like your answer better.

 

[Dick]

Absolutely, now what is your answer? I'd like to see it to make sure we're on the same wavelength.

 

[KyleS4562]

A. Invertible. Let's call this transformation T which as a transformation matrix A. So since it is invertible there exists an A^-1 such that: T(<x,y,z>)= <x1,y1,z1> where A<x,y,z> = <x1,y1,z1> so that A^-1 <x1,y1,z1> = <x,y,z> for any given plane

and something like that for the other ones or am I being to vague?

 

[Dick]

No, you are being too specific. Isn't the inverse of a reflection the same reflection? Just say that, I know you know it.

 

[KyleS4562]

Wow I didn't think of that. So it is just A^-1 = A

then for b we can actually do it since

A= [5,0,0] [0,5,0] [0,0,5]

A^-1 = [1/5,0,0] [0,1/5,0] [0,0,1/5]

then d if a rotation around an axis is defined by theta degrees than the inverse matrix will rotate the vector 2pi-theta in the same direction

 

[Dick]

Right, exactly. The inverse of scaling by 5 is scaling by 1/5. And the inverse of a rotation around an axis by theta is rotation around the same axis by -theta. Or as you put it 2pi-theta. But they are the same thing.

 

[KyleS4562]

Thank you very much for all your help