Linear Algebra

Hi, just to ask for a bit of help here, not too sure what to do...

Vector space in R3, {(x,y,z) | x,y,z are in R}, with set S = {(x,y,z) in R3 | x + y - z = 0} being a vector space over R.

Consider v = (-1,0,-1) and w = (-1,1,0)

Prove that every vector s in S can be written as a linear combination of v and w and hence that v and w form a basis for S.

I have no real clue how to go about this, anything i've tried doesn't work so any help would be really appreciated! Thanks

Prove that every vector s in S can be written as a linear combination of v and w and hence that v and w form a basis for S.

Hi, just to ask for a bit of help here, not too sure what to do...

Vector space in R3, {(x,y,z) | x,y,z are in R}, with set S = {(x,y,z) in R3 | x + y - z = 0} being a vector space over R.

Consider v = (-1,0,-1) and w = (-1,1,0)

Prove that every vector s in S can be written as a linear combination of v and w and hence that v and w form a basis for S.

I have no real clue how to go about this, anything i've tried doesn't work so any help would be really appreciated! Thanks

another way to go about it is note that for s ∈ S, s=(x,y,x+y) = x(1,0,1) +y(0,1,1)

So s ∈ Sp({(1,0,1), (0,1,1)})

Can you apply the exchange lemma here?

Edit: Firegardens method is pretty much the same

More concretely, being in S entails that the vector has the form (x, y, x+y) since z=x+y. Can you find a combination of v and w such that for any given x and y, you get the vector (x, y, x+y)? If so, every vector in S can be written as a linear combination of v and w.

Then you need to prove linear independence of v and w to prove they form a basis rather than just a spanning set.

Thank you all very much but i've been sitting here for ages trying to find a combination and can't work one out, is there a combination? This subject gives me headaches haha

Thank you all very much but i've been sitting here for ages trying to find a combination and can't work one out, is there a combination? This subject gives me headaches haha

v= -1(1,0,1)
I'm sure you can work out what w should be (after doing the first exchange)

Actually thinking about it, what do you mean by combination?

If you find LA hard (I did as well), I would recommend getting a couple of textbooks out of the library, I'll give you a list of ones I found useful in a bit.

I feel like a massive idiot right now haha, i've been trying to work this out like

c1v + c2v = (x,y,x+y)

but any combination i get won't work out, i thought it was v=-1(1,0,1) but I couldn't work out a term for c2

(I've never heard of the exchange lemma either)

https://www0.maths.ox.ac.uk/system/files/coursematerial/2015/2631/55/LAnotes-14.pdf

Page 25. But I wouldn't worry about learning it if you don't know it, I don't know what you do in your LA class so don't know what theorems you should be applying to this question

Good luck for your exam!

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