Is there a Linear Transformation

1. Jan 19, 2015

Chillguy

1. The problem statement, all variables and given/known data
From Hoffman and Kunze:

Is there a linear transformation T from $$R^3$$ to $$R^2$$ such that T(1,-1,1)=(1,0) and T(1,1,1)=(0,1)?

2. Relevant equations
$$T(c\alpha+\beta)=cT(\alpha)+T(\beta)$$

3. The attempt at a solution
I don't really understand how to prove that there is a linear transformation with these coordinates. I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1). Then I don't really know where to go from here.

2. Jan 19, 2015

Orodruin

Staff Emeritus
Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?

3. Jan 19, 2015

Chillguy

It would allow me to span R3 with these three vectors

4. Jan 19, 2015

Orodruin

Staff Emeritus
And thus the transformation of any vector in R3 can be written as ...

5. Jan 19, 2015

Chillguy

... can be written as $$T(c\alpha_i+\beta_i)?$$
where alpha and beta are vectors in R3

6. Jan 19, 2015

Orodruin

Staff Emeritus
Well, you need to define what your $\alpha$ and $\beta$ etc are. You have three vectors with which you can express any vector in R3 as a linear combination. What will be the resulting transformation of such a linear combination? Will it fulfil the requirements you have?

7. Jan 19, 2015

Chillguy

The resulting combination will be a vector in R^2, I think it could be any vector in R^2?

8. Jan 19, 2015

Orodruin

Staff Emeritus
No, not if you already have fixed it for three vectors. Use the linear property of T! Let us say you have the vectors $v_1, v_2, v_3$ and have fixed $T(v_i) = u_i$ for $i = 1,2,3$. Now you take a linear combination of those $v = \sum_i c_i v_i$. What is $T(v)$?

9. Jan 19, 2015

Chillguy

$$T(v)=T(c*v_1+c*v_2+c*v_3)$$
Which I can then apply the definition of a linear transformation as defined before?

10. Jan 19, 2015

Orodruin

Staff Emeritus
Yes. So does this linear transformation fulfil the requirements?

11. Jan 19, 2015

Chillguy

Yes!!! it does! Thank you so much. It makes much more sense now. I just have one other question, why do the vectors need to span all of R^3 in the first place? Is it so the transformation can go to any place in R^2?

12. Jan 19, 2015

Orodruin

Staff Emeritus
Unless you specify the transformation for any arbitrary vector in R3 you have not really found a linear transformation from R3 to R2 but only from a two-dimensional subspace. A more complete answer would be: Yes, it exists, but is not unique. (You could have selected any vector in R2 as the image of your third linearly independent vector in R3).