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## gorica 2 years ago 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

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1. terenzreignz

What does S have to do with this though?

2. gorica

sorry :D W is generated by S :)

3. terenzreignz

meaning... S spans W?

4. gorica

yes

5. terenzreignz

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.

6. gorica

I know, but I don't know how

7. terenzreignz

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...

8. gorica

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?

9. terenzreignz

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.

10. gorica

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

11. terenzreignz

You can do it from here?

12. gorica

you mean to write it here?

13. terenzreignz

I'm not even finished thinking about it XD

14. terenzreignz

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

15. gorica

I'll write it :)

16. terenzreignz

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

17. gorica

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

18. terenzreignz

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

19. gorica

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 :)

20. terenzreignz

No problem :)

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