I am having difficulties with the following problem. I need to find values of t that make the set linearly independent.
Problem:
The answer is all t not equal to 1 and -2. I am confused on how to approach this problem.
If I set up a matrix I believe I see where -2 comes from. And we need t to not equal that otherwise that's another solution to the homogeneous equation. Matrix:
Beyond seeing where those two numbers come from, I am not sure of the procedure behind getting them. Thanks for any help.
If I try bringing it to RREF, I get a jumble for a matrix. Taking your advice:
If t = 1 here I get the trivial solution as everything becomes zero. I still think I am missing something. Not seeing where the -2 is coming from regarding the above matrix. What I have done seems to imply t = 2. Thanks for your patience.
I would be inclined to go back to the basic definition of "linearly independent": for what t does a(t,1,1)+ b(1,t,1)+ c(1,1,t)= (0,0,0) have only the solution a= b= c= 0? That gives the three equations at+b+c= 0, a+bt+ c= 0, and a+ b+ ct= 0 (Which, of course have exactly the coefficient matrix you use). If we subtract the second equation from the first, we get (t-1)a+ (1-t)b= 0 or (t-1)a= (t-1)b so a= b for all t except t= 1. Similarly, subtracting the third matrix from the first gives (t- 1)a+ (1- t)c= 0 or (t-1)a= (t-1)c so a= c for all t except 1. Putting b=a and c= a into the first equation, ta+ a+ a= (t+2)a= 0 and we must have a= b= 0 unless t= -2.