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 2 years 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
 2 years 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

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terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1What does S have to do with this though?

gorica
 2 years ago
Best ResponseYou've already chosen the best response.0sorry :D W is generated by S :)

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1meaning... S spans W?

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1Well, 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
 2 years ago
Best ResponseYou've already chosen the best response.0I know, but I don't know how

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1Unfortunately, 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
 2 years ago
Best ResponseYou've already chosen the best response.0should 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
 2 years ago
Best ResponseYou've already chosen the best response.1It 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
 2 years ago
Best ResponseYou've already chosen the best response.0ok, thanks :) that's what was confusing me :)

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1You can do it from here?

gorica
 2 years ago
Best ResponseYou've already chosen the best response.0you mean to write it here?

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1I'm not even finished thinking about it XD

terenzreignz
 2 years ago
Best ResponseYou've already chosen the best response.1// too comfy to stand up and look for a pencil and paper XD

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

gorica
 2 years ago
Best ResponseYou've already chosen the best response.0but 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
 2 years ago
Best ResponseYou've already chosen the best response.1No... 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
 2 years ago
Best ResponseYou've already chosen the best response.0yes, 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 :)
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