# Not Satisfied with Linear Algebra Theorem

by Vorde
Tags: algebra, linear, satisfied, theorem
 P: 784 Hello, I am just not satisfied with the following theorem (I don't know it's name): Let T:R^n -> R^m be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution. The "proof" involves saying that if T is not one-to-one, then there are two different vectors U and V such that T(U)=T(V)= some vector B. And since T is linear it follows that T(U-V) = T(U)-T(V) = B - B = 0. It then concludes by saying "hence there are nontrivial solutions to T(X)= 0. So, either the two conditions in the theorem are both true or they are both false." I just don't see how that proved the theorem in any way, perhaps because I don't fully understand which two conditions it is talking about. Could anyone help me here? Thank you.
 P: 4,575 Hey Vorde. A 1-1 would imply that T(U-V) = T(U) - T(V) != 0 for U != V. If however this were false then it would imply that for some U != V that T(U) = T(V) proving that the mapping is not 1-1. But then you have to take into account the trivial solution (i.e. the zero vector) as a special case where T(U) = 0 for U = 0. The formal proof for 1-1-ness is to show that for U != V then T(U) != T(V) for all U and V in the domain of the mapping.
 P: 784 Hey Chiro, Right, so I understand that part. What I don't understand is how it is enough to show that there aren't any non-zero vectors that map to the zero-vector to know that the mapping is one-to-one everywhere. Why couldn't the zero vector be the only vector that maps to the zero vector but still have non 1-1-ness elsewhere?
 P: 4,575 Not Satisfied with Linear Algebra Theorem We know that the zero vector always maps to the zero vector, but we also know that if everything is 1-1, then it means that only the zero vector maps to the zero vector and everything else maps to some other vector (that isn't the zero vector).
P: 838
 Quote by Vorde Why couldn't the zero vector be the only vector that maps to the zero vector but still have non 1-1-ness elsewhere?
Take two vectors, u and v.
Suppose T(u) = T(v).
Then T(u) - T(v) = 0.
But be linearity, T(u) - T(v) = T(u-v).
Then only vector that maps to 0 is the zero vector.
Hence, u - v = 0, so u = v, proving T was 1-to-1.
 P: 784 Okay, but that only works if you posit that the transformation is 1-1, and the theorem doesn't start with the assumption that the transformation is 1-1. I understand that the zero vector will always map to itself, just not why that says anything about the rest of the transformation. Ah, wait, I think I might see it now.
 C. Spirit Sci Advisor Thanks P: 5,661 Let $T$ be injective (sorry I absolutely hate the term 1 - 1 god knows why it is even used). Because T is linear, $T(0) = 0$. This immediately implies this is the only vector for which this is true because if $\exists v\in V :T(v) = 0$, then the injectivity of $T$ implies that $v = 0$. Now let $T(v) = 0$ be true only for the zero element then if $T(v) = T(w)$ we have that $T(v - w) = 0\Rightarrow v = w$ thus $T$ is injective. This is essentially what the proof you quoted is saying but the quoted proof is more concise. EDIT: Seems like people responded while I was typing this up but I guess I'll leave it here anyways =D.
 P: 784 Okay, thank you to all who have been helping. What was really bothering me was that I didn't see why the trivial solution had to be the only solution for the zero vector. But I just went back and thought about it and now I can see that this must be the case. That assumed, I can follow the rest of the theorem. Once again, thank you all.
 HW Helper P: 2,264 T is one to one if whenever T(u)=T(v) we must have u=v now suppose T(x)=0 only when x=0 If T(u)=T(v) by linearity T(u-v)=0 then we must have u-v=0 so u=v

 Related Discussions Linear & Abstract Algebra 2 Linear & Abstract Algebra 10 Calculus & Beyond Homework 8 Linear & Abstract Algebra 3 Calculus & Beyond Homework 2