Question

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.]

Answer 1

Answer:

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 $c_{1} T(v_{1} ) + c_{2} T(v_{2} ) + c_{3} T(v_{3}) = 0$

Let $c_{i}$ be a set of non-zero numbers such that that $c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3} = 0$

Because T is a linear solution:

$c_{1}T(v_{1} ) + c_{2} T (v_{2} ) + c_{3} T (v_{3} ) = T (c_{1} v_{1} ) + T(c_{2}v_{2} ) + T(c_{3} v_{3} )$

                                            = $T(c_{1}v_{1} + c_{2} v_{2} + c_{3} v_{3} )$

                                            = $T(0)$

                                            = $T (00)$

                                            = $0T(0)$

                                            = 0

This shows that $T( v_{1} )$ are linearly independent with the mapping resulting in a non-trivial solution.