# Determine whether or not T is one-to-one in each of the following situations?

0

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!!

• 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.

• 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.

Also Check This  What is the electric potential energy of the group of charges in the figure? (Figure 1) Express your answer wi? 