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 dependant 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.)

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