Let T be a linear transformation from R^r to R^s .
Determine whether or not T is one-to-one in each of the following situations:
r=s
r>s
r<s
Explanation would be appreciated!!
2 Answers
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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 one-to-one. In this case, it will be one-to-one 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 one-to-one.
If r > s, I can not be one-to-one. 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 non-zero in R^r such that
T(x) = T(0)….two inputs-one output–not 1:1
If r < s, T may be one-to-one. It can’t be onto.
In none of the three cases are we guaranteed that T is one-to-one (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.
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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 1-1? 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 1-1? Still by looking as concrete examples you can see the general pattern.
