# Homework Help: Matrix Linear Systems

1. May 7, 2014

### B18

1. The problem statement, all variables and given/known data
We know that X= (1 over 3)e^t+(4 over -4) te^t is a solution to X'=(2 1 over -1 0)X.

Verify that the solution vectors are linearly independent on (-∞,∞).

2. Relevant equations
I know that the wronskian of the solution vectors cannot be 0 if they are linearly independent.

3. The attempt at a solution
So I found X1=(1 over 3) e^t and X2= (4 over -4)te^t
when i did the wronskian i did..
l e^t 4te^t l
l 3e^t -4te^t l

I got -16te^(2t). this would be 0 at t=0. Am i missing something? i figured that it should be independent because it says to verify that it is... are these linearly dependent?

thanks to any one for the help!

2. May 7, 2014

### Dick

If the wronskian is nonzero ANYWHERE then the functions are linearly independent. If they are dependent it will vanish everywhere.

3. May 7, 2014

### B18

Okay, so if the wronskian is equal to 0 and only 0 then the solutions are linearly dependent. However if the wronskian is anything other than 0 they are linearly independent!

4. May 7, 2014

### Dick

Yes, they are linearly independent.