# Please help solving Linearly Dependent question for 4 variation

• Nov 2nd 2012, 07:46 AM
angelme
cosx, sinx, ex, and e-x are LI or LD ( 0 =or< x =or< 1)

I can see without calculation that it is LI but I do not know how to prove it.

I tried to use Wronskian Determinant but it gets really messy

W(x) =

cosx sinx ex e-x
-sinx cosx ex -ex

-cosx -sinx ex e-x
sinx -cosx ex e-x

Someone please suggest a better and quicker way...(Headbang)

Thank you :(
• Nov 3rd 2012, 02:38 PM
GJA
Re: Please help solving Linearly Dependent question for 4 variation
Hi angelme,

I will call the Wronskian matrix \$\displaystyle W.\$ To start, we should have \$\displaystyle W_{4,4}=-e^{-x};\$ in other words the entry in the fourth row, fourth column should be \$\displaystyle -e^{-x},\$ not \$\displaystyle e^{-x}.\$ I'm guessing you actually have this and missed a minus sign in your post.

To make life simpler, using the fact that the determinant is not affected when you add a row to another will make things easier. For example, adding row 1 to row 3 makes row 3 become \$\displaystyle 0, 0, 2 e^{x}, 2e^{-x}.\$ If you use this fact a couple of times you should be able to get a row that has 3 zeros and one nonzero entry. You then expand your determinant about this nonzero entry and go from there.

Give it another shot with this in mind. I ended up getting \$\displaystyle det(W)=-6\$ when all was said and done. Since \$\displaystyle -6\neq 0\$ the vectors are linearly independent. Let me know if you still get stuck. Good luck!
• Nov 7th 2012, 07:20 PM
angelme
Re: Please help solving Linearly Dependent question for 4 variation
sorry I'm still not getting it :( Could you please explain me step by step?