Check linear independence of the functions 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.
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.