Let T W Rn ! Rm be a linear transformation. Show that if T maps two linearly independent vectors onto a linearly dependent set, then the equation T .x/ D 0 has a nontrivial solution. [Hint: Suppose u and v in Rn are linearly inde- pendent and yet T .u/ and T .v/ are linearly dependent. Then 495 c1T .u/ C c2T .v/ D 0 for some weights c1 and c2, not both zero. Use this equation.]

The solution proves that the equation has a non-trivial solution.  

Step-by-step explanation:

We want to show that the equation has nontrivial solutions for which

Let be a set of non-zero numbers such that that

Because T is a linear solution:

                                            =

                                            =

                                            =

                                            =

                                            = 0

This shows that are linearly independent with the mapping resulting in a non-trivial solution.

21.4 i did it already for s1

step-by-step explanation: