1. Linear transformation question

Is there a linear transformation T : R3 -> R2 such that T(1, -1, 1) = (1, 0) and T(1, 1, 1) = (0, 1)? Justify your answer.

I tried using the definition of a linear map to make a contradiction so to prove there is not a linear transformation that meets the above requirements. But I don't think I proved anything that way. For all I know, there could be a linear transformation that meets the above requirements but I don't know how to find it.

2. Originally Posted by Zalren
Is there a linear transformation T : R3 -> R2 such that T(1, -1, 1) = (1, 0) and T(1, 1, 1) = (0, 1)? Justify your answer.

I tried using the definition of a linear map to make a contradiction so to prove there is not a linear transformation that meets the above requirements. But I don't think I proved anything that way. For all I know, there could be a linear transformation that meets the above requirements but I don't know how to find it.
The difficulty is that there is such a linear transformation! The two vectors (1, -1, 1) and (1, 1, 1) are independent- you can map them into anything. Further, (0, 0, 1) is independent of both so {(1, -1, 1), (1, 1, 1), (0, 0, 1)} is a basis for $\displaystyle R^3$. Define T(0, 0, 1) to be whatever you like, say, T(0, 0, 1)= (0, 0), and, together with the first two equations, you have defined a linear transformation from $\displaystyle R^3$ to $\displaystyle R^2$ such that T(1, -1, 1)= (1, 0) and T(1, 1, 1)= (0,1)

3. Originally Posted by HallsofIvy
The difficulty is that there is such a linear transformation! The two vectors (1, -1, 1) and (1, 1, 1) are independent- you can map them into anything. Further, (0, 0, 1) is independent of both so {(1, -1, 1), (1, 1, 1), (0, 0, 1)} is a basis for $\displaystyle R^3$. Define T(0, 0, 1) to be whatever you like, say, T(0, 0, 1)= (0, 0), and, together with the first two equations, you have defined a linear transformation from $\displaystyle R^3$ to $\displaystyle R^2$ such that T(1, -1, 1)= (1, 0) and T(1, 1, 1)= (0,1)
Thank you so much!