Linear transformation T: R3 --> R2

[aero_zeppelin]

1. The problem statement, all variables and given/known data

Find the linear transformation T: R3 --> R2 such that:
T(1,0,0) = (2,1)
T(0,1,1) = (3,2)
T(1,1,0) = (1,4)



3. The attempt at a solution

I've been doing some exercises about linear transformations (rotations and reflections mostly) but I've never seen something like this... I don't know how to even start :S I did some research and only found examples on proving the two conditions for a transformation to be linear...

Any help please?

[Dick]

Could you answer the problem if they gave you the values of T(1,0,0), T(0,1,0) and T(0,0,1)? If you can than can you figure out what they are from the given information?

[aero_zeppelin]

Thanks for the reply.

Humm.. what do you mean? If they changed the values of the T's I posted?

[Dick]

Thanks for the reply.

Humm.. what do you mean? If they changed the values of the T's I posted?

Yes, T is a 2x3 matrix. T(1,0,0) is the first column of the matrix, right?

[aero_zeppelin]

Yes, T is a 2x3 matrix. T(1,0,0) is the first column of the matrix, right?

Yeah, I would say so... Like I said, I have never done an exercise like this before, so I don't exactly now how to start arranging things hehe

[Dick]

Yeah, I would say so... Like I said, I have never done an exercise like this before, so I don't exactly now how to start arranging things hehe

Well, it's time to start arranging things now. You know T(1,0,0). They gave it to you. So you know the first column of the matrix. Now what's T(0,1,0)? That would be the second column. Can you figure that out using linearity from the given information?

[aero_zeppelin]

Well, it's time to start arranging things now. You know T(1,0,0). They gave it to you. So you know the first column of the matrix. Now what's T(0,1,0)? That would be the second column. Can you figure that out using linearity from the given information?


hehe ok! Are you mistaking that second T? It should be T(0,1,1), right? Anyway, what exactly are we trying to do? Arrange T's inside a matrix as columns and then...?

[HallsofIvy]

Another way to do this: Write <x, y, z> as a combination of <1, 0, 0>, <0, 1, 1>, and <1, 1, 0>. That is, <x, y, z>= a<1, 0, 0>+ b<0, 1, 1>+ c<1, 1, 0>= <a+ c, b+ c, b>. So we have a+ c= x, b+ c= y, and b= z. Then c= y- z and a= x- y+ z.

That is, <x, y, z>= (x-y+ z)<1, 0, 0>+ z<0, 1, 1>+ (y- z)<1, 1, 0>.

So T<x, y, z>= (x- y+ z)T<1, 0, 0>+ zT<0, 1, 1>+ (y- z)T<1, 1, 0>.

[Dick]

hehe ok! Are you mistaking that second T? It should be T(0,1,1), right? Anyway, what exactly are we trying to do? Arrange T's inside a matrix as columns and then...?

What is T(1,1,0)-T(1,0,0)? In the example they gave it is easy to find the value of T on the three basis vectors (1,0,0), (0,1,0) and (0,0,1). That will let you write down a matrix for T. I was guessing that was what they meant by 'find the linear transformation'.