Suppose that T is a one-to-one transformation, so that an equation T(u)=T(v) always implies u=v. Show that if the set of images {T(v1)T(vp)} is linearly dependent, then {v1vp} is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).

Step-by-step explanation:

The objective is to  Show that if the set of images {T(v1)......T(vp)} is linearly dependent, then {v1......vp} is linearly dependent.

Given that:

is linearly dependent set

Thus; there exists scalars ; ( read as "such that")  

T = 0                                  (for the fact that T is linear transformation)

    (due to T is one-one)

NOTE: Not all Ki's are zero;

Thus;

 is linearly dependent

It negation also illustrates that :

If is also linearly independent then is also linearly independent.

correct

step-by-step explanation: