# Homework Help: 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).

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