Let S={(1,k,k+1,k-1)|k is real} and W is subspace of vector space R4. Prove that a=(0,1,1,1) is in W

Question

terenzreignz · Answer

What does S have to do with this though?

gorica · Answer

sorry :D W is generated by S :)

terenzreignz · Answer

meaning... S spans W?

gorica · Answer

yes

terenzreignz · Answer

Well, then, all you need to show, for the meantime, is that (0,1,1,1) can be written as a linear combination of elements in S.

gorica · Answer

I know, but I don't know how

terenzreignz · Answer

Unfortunately, I know no other way than to figure out exactly what linear combination of S could possibly be equal to (0,1,1,1).
Stand by...

gorica · Answer

should I take vectors from S for different k and make linear combination... for example
(0,1,1,1)=a*(1,2,3,1)+b*(1,1,2,0)+... how many vectors should I take? 4?

terenzreignz · Answer

It can be as many vectors as you like, but there may well be a minimum.  Yes, you form vectors from S using different values for k.

gorica · Answer

ok, thanks :) that's what was confusing me :)

terenzreignz · Answer

You can do it from here?

gorica · Answer

you mean to write it here?

terenzreignz · Answer

I'm not even finished thinking about it XD

terenzreignz · Answer

// too comfy to stand up and look for a pencil and paper XD

gorica · Answer

I'll write it :)

terenzreignz · Answer

Never mind, it's actually a lot simpler than I thought... just consider the elements of S when k=0 and k=1.

gorica · Answer

but I think I can't take less than 4 vectors from S since dimension of R4 is 4, so I have to write (0,1,1,1) as linear combination of 4 vectors

terenzreignz · Answer

No... the dimension of R4 is 4, so that means any vector can be written as a linear combination of 4 vectors, it doesn't mean that it can't be smaller than that.

For instance, in R4 itself, the vector (1,0,0,0) can certainly be written as a linear combination of four vectors:
(1,-1,0,0) + (0,1,-1,0)+(0,0,1,-1)+(0,0,0,1)
But it doesn't mean it can't be written as a linear combination of two:
(1,-1,0,0) + (0,1,0,0)
Or even just one
(1,0,0,0)

Get the drift? XD

gorica · Answer

yes, but... it's safer to write with 4 vectors and get some 0 coefficients :D 
never mind... I go now, have exam at 5 :) wish me luck :)
and thanks again :)

terenzreignz · Answer

No problem :)