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Let S={(1,k,k+1,k1)k is real} and W is subspace of vector space R4. Prove that a=(0,1,1,1) is in W
 one year ago
 one year ago
Let S={(1,k,k+1,k1)k is real} and W is subspace of vector space R4. Prove that a=(0,1,1,1) is in W
 one year ago
 one year ago

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terenzreignzBest ResponseYou've already chosen the best response.1
What does S have to do with this though?
 one year ago

goricaBest ResponseYou've already chosen the best response.0
sorry :D W is generated by S :)
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
meaning... S spans W?
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
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.
 one year ago

goricaBest ResponseYou've already chosen the best response.0
I know, but I don't know how
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
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...
 one year ago

goricaBest ResponseYou've already chosen the best response.0
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?
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
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.
 one year ago

goricaBest ResponseYou've already chosen the best response.0
ok, thanks :) that's what was confusing me :)
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
You can do it from here?
 one year ago

goricaBest ResponseYou've already chosen the best response.0
you mean to write it here?
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
I'm not even finished thinking about it XD
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
// too comfy to stand up and look for a pencil and paper XD
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
Never mind, it's actually a lot simpler than I thought... just consider the elements of S when k=0 and k=1.
 one year ago

goricaBest ResponseYou've already chosen the best response.0
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
 one year ago

terenzreignzBest ResponseYou've already chosen the best response.1
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
 one year ago

goricaBest ResponseYou've already chosen the best response.0
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 :)
 one year ago
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