Element of a vector group (subset?)

[Mewniew]

EDIT: I think i screwed up and posted in the wrong section. Sorry. Should i make a new one to the correct place? Can this one be moved?

Hello. I have a couple of problems here, that i will have to translate from another language, so I am not 100% sure if I am using the correct terms.

(1) Let u, v, w ∈ ℝⁿ be vectors. Find out if the vector u+v is an element of the group (subset?) (2u-v, 2u-w, 2w-u).

(2) Show that the ℝⁿ vectors u=(a,1,-1), v=(b,-1,2) and w=(c,1,3) are linearly dependent exactly when c-4b-5a=0.

So, i don't know what to do with these.

 

[Svein]

In (2) it seems to me that you are talking about vectors in ℝ³. This observation should make that problem easier.

In (1) you must check if there are scalars a, b and c such that a⋅(2u-v)+b⋅(2u-w)+c⋅(2w-u)=u+v.

 

[Mewniew]

In (2) it seems to me that you are talking about vectors in ℝ³. This observation should make that problem easier.

In (1) you must check if there are scalars a, b and c such that a⋅(2u-v)+b⋅(2u-w)+c⋅(2w-u)=u+v.

Thank you for answering! For (2), that is a typo, it's supposed to say ℝ³. I still don't quite understand what I am supposed to do.

 

[Svein]

For (2), that is a typo, it's supposed to say ℝ3. I still don't quite understand what I am supposed to do.

Again, it is a question regarding a set of equations. You have three equations with three unknowns - for what values of a, b , c is there no solution?

 

[Mewniew]

Again, it is a question regarding a set of equations. You have three equations with three unknowns - for what values of a, b , c is there no solution?

Dont i have 4 equations and 6 unknowns? The scalars and a,b and c?

 

[Svein]

Have you heard of "Cramer's Rule"? If not, check out http://www.purplemath.com/modules/cramers.htm.

 

[Mewniew]

I hadn't heard of it, and i haven't learned determinant yet, either :F

 

[Mewniew]

So here are my attempts:
(1) https://dl.dropboxusercontent.com/u/34732003/h/20151011_220232.jpg
I stopped, because i wasnt sure what i was doing and it felt like nonsense.

(2) https://dl.dropboxusercontent.com/u/34732003/h/20151011_220130.jpg
Numbers show the order in which i did things. 1 and 2 seemed like things i could do, then i started the matrix and realized i had no idea what to do with the a, b and c. Then, i moved on to 4 to see if i anything new and interesting showed up by mixing and matching the equations. It didnt. (Now i realize they're not all supposed to equal 0, as they are L.dependent.)

 

[Mark44]

So here are my attempts:
(1) https://dl.dropboxusercontent.com/u/34732003/h/20151011_220232.jpg
I stopped, because i wasnt sure what i was doing and it felt like nonsense.

(2) https://dl.dropboxusercontent.com/u/34732003/h/20151011_220130.jpg
Numbers show the order in which i did things. 1 and 2 seemed like things i could do, then i started the matrix and realized i had no idea what to do with the a, b and c. Then, i moved on to 4 to see if i anything new and interesting showed up by mixing and matching the equations. It didnt. (Now i realize they're not all supposed to equal 0, as they are L.dependent.)

Please post your attempts here, rather than as images on dropbox. Everything you have done can be done using either BBcodes or LaTeX. We have tutorials on each of these types of markup.
LaTeX: https://www.physicsforums.com/help/latexhelp/
BBcodes: https://www.physicsforums.com/help/bb-codes

Both can be accessed from INFO --> Help/How-To