Let T be a linear transformation from R^r to R^s .
Determine whether or not T is onetoone in each of the following situations:
r=s
r>s
r<s
Explanation would be appreciated!!
2 Answers

There is a theorem that you are probably familiar with. The dimension of the range of T plus the dimension of the kernel of T is equal to r.
If r = s, then T may be onetoone. In this case, it will be onetoone if and only if it is onto. To know, you have to determine what the kernel is. If the kernel of T is {0}, then it is onetoone.
If r > s, I can not be onetoone. The largest the dimension of the image could be is s (in the case that T is onto). So the dimension of the kernel is at least r – s >0. This means that there are nonzero vectors that map to the zero in R^s. That is, there exist x nonzero in R^r such that
T(x) = T(0)….two inputsone output–not 1:1
If r < s, T may be onetoone. It can’t be onto.
In none of the three cases are we guaranteed that T is onetoone (take for example the transformatin T(x)=0 for all x which can be a linear transformation in all three cases.) But in the first and third, it is possible to have a 1:1 mapping.

Rather than work in the abstract, sometimes it helps to try a simpler problem to build your intuition.
So let r = 2 and s = 2.
If T is linear transformation from the plane to the plane, need it be 11? Nope Set T(x,y) = 0.
You can work out the others. But you may have missed something in the question.
Maybe the question was: Can T be 11? Still by looking as concrete examples you can see the general pattern.