Let Txrightarrow Axrightarrow be a linear transformation wh

Let T(x^rightarrow) = Ax^rightarrow be a linear transformation where A is a m times n matrix. If n m, can the mapping T be one-to-one? If it can, does it have to be?

Solution

We know that when T: RⁿR^m is a linear transformation,the following are equivalent:

Also, when T: RⁿR^m is a linear transformation,the following are equivalent:

Now we come to the question:

a. If n < m, then rank(A) n < m so that T cannot be onto.

b. If n< m, T can be one-to-one if Rank (A) = n.

c. If n > m, then T can be onto . T will be onto when rank(A) = m.

d. If n > m, then rank(A) m < n so that T cannot be one-to-one.