Is there a Linear Transformation

[Chillguy]

Homework Statement

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)?

Homework Equations

$$T(c\alpha+\beta)=cT(\alpha)+T(\beta)$$

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.

 

[Orodruin]

I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1).

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?

 

[Chillguy]

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?

It would allow me to span R³ with these three vectors

 

[Orodruin]

It would allow me to span R³ with these three vectors

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

 

[Chillguy]

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

 

[Orodruin]

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?

 

[Chillguy]

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

 

[Orodruin]

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 $T(v)$?

 

[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?

 

[Orodruin]

Yes. So does this linear transformation fulfil the requirements?

 

[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?

 

[Orodruin]

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).