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 [tex]c_{1} T(v_{1} ) + c_{2} T(v_{2} ) + c_{3} T(v_{3}) = 0[/tex]
Let [tex]c_{i}[/tex] be a set of non-zero numbers such that that [tex]c_{1} v_{1} + c_{2} v_{2} + c_{3} v_{3} = 0[/tex]
Because T is a linear solution:
[tex]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} )[/tex]
= [tex]T(c_{1}v_{1} + c_{2} v_{2} + c_{3} v_{3} )[/tex]
= [tex]T(0)[/tex]
= [tex]T (00)[/tex]
= [tex]0T(0)[/tex]
= 0
This shows that [tex]T( v_{1} )[/tex] are linearly independent with the mapping resulting in a non-trivial solution.
