I'm learning differential equations recently.Our teacher gave some exercies for us, I did almost all of them but I couldn't understand how to check linear independence , actually I have studied some examples but this questions are looks different for me.
If anybody can show way or any example about them i will be so happy, it's urgent for me.
Check linear independence of the function :
Check that the functions are linearly independent solutions of the system. Find the general solution of the system and a fundamental matrix.
thank you :)
If $\displaystyle x(t) = \left (\array{x_1(t) \\ x_2(t)} \right) \hbox{and } y(t) = \left (\array{y_1(t) \\ y_2(t)} \right )$
then take the cross-product of the two functions, which is: $x_1(t)y_2(t) - x_2(t)y_1(t)$
If this is zero, the functions are dependent. If it is non-zero, the functions are independent.