Linear algebra ordinary differential equations

[SpiffyEh]

Homework Statement

I attached the problem in a picture, I'm not so good at making it show correctly on here.

Homework Equations

The Attempt at a Solution

I'm really unsure of what to do with this problem. I don't know where to start. Can someone please try to guide me through it so I understand? Thank you

 

[Dick]

Either you chopped off part of the problem or the typesetters of your book did. What you've quoted doesn't tell you what they want you to show about V. I suspect they want you to show V is a vector space. Does that help?

 

[SpiffyEh]

Thats all that was given for that problem. I really don't know what to do

 

[Dick]

Thats all that was given for that problem. I really don't know what to do

If that's all there is then the book is typographically flawed. That's not even a question. It ends with a ',' where the question ought to be. You have two options i) you can complain about it and try to get out of trying to solve it or ii) you can guess what the problem should be (which is pretty obvious from the hint) and try to show V is a vector space.

 

[SpiffyEh]

well, I don't think I'm going to get out of it. It was typed by the professor, his questions tend to be confusing. I think ii would be my own option. How would I go about showing V is a vector space?

 

[Dick]

Use the hint. Show if L=0 and L[v]=0 then L[c1*u+c2*v]=0 where c1 and c2 are constants.

 

[SpiffyEh]

I don't know how to do that, that's why I'm having issues with the problem.

 

[Dick]

I don't know how to do that, that's why I'm having issues with the problem.

That's not good. Do you see why if L[y]=0 then L[c*y]=0, where c is a constant? Can't you factor a constant out of all of the derivatives?

 

[SpiffyEh]

I understand that part about how I can factor things out but I don't understand how to show that u,v are in V so that I can show the part with the constants.

 

[Dick]

I understand that part about how I can factor things out but I don't understand how to show that u,v are in V so that I can show the part with the constants.

You don't know why L[u+v]=L+L[v]? If D is one your derivatives isn't D[u+v]=D+D[v]??

 

[SpiffyEh]

I guess I was reading it wrong. I thought I had to prove that u,v are in V but since they are L = 0 amd L[v] = 0 so I can split it up like that and bring out the constants. I understand how it can be split up. But does the hint actually prove the point of the problem? Or is it too unclear to see?

 

[Dick]

You are given u and v are in V. That means L=0 and L[v]=0. If you are clear on why L[c1*u+c2*v]=0 meaning c1*u+c2*v is also in V then that's the whole problem. It means V is a vector space.