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 November 17th, 2011, 10:02 AM #1 Newbie   Joined: Nov 2011 Posts: 2 Thanks: 0 Vector spaces Can you help me with this problem, please? If these two operations are defined as follows. With these new definitions, is R3 a vector space? a) (x1, y1, z1)+(x2, y2, z2)=(x1 + x2 + 2, y1 + y2 + 2, z1 + z2 + 2) c(x, y, z)=(cx, cy, cz) b) (x1, y1, z1)+(x2, y2, z2)=(x1 + x2 + 4, y1 + y2 + 4, z1 + z2 + 4) c(x, y, z)=(cx + 4c – 4, cy + 4c – 4, cz + 4c – 4) Thanks!
 November 18th, 2011, 10:19 AM #2 Member   Joined: Jun 2010 Posts: 64 Thanks: 0 Re: Vector spaces a) (x1, y1, z1)+(x2, y2, z2)=(x1 + x2 + 2, y1 + y2 + 2, z1 + z2 + 2) c(x, y, z)=(cx, cy, cz) b) (x1, y1, z1)+(x2, y2, z2)=(x1 + x2 + 4, y1 + y2 + 4, z1 + z2 + 4) c(x, y, z)=(cx + 4c – 4, cy + 4c – 4, cz + 4c – 4) __________ With this operations the (R^3+) is not group Suppose that (a,b,c) be the zero elemet then (a,b,c)+(a,b,c)=(a,b,c) from this we have taht a+a+2=a then a=-2 similar we have zero element o=(-2,-2,-2), By the other operation if we have suppose that R^3 is vector space than we would have that for every c( get c=4) c*0=0 for c=4 c*0=4(-2,-2,-2)=(-8,-8,-=(-2,-2,-2) absurde !!! b) similary find the zero 0, suppose 0=(x1,x2,x3) then 0+0=0 (x1, y1, z1)+(x1, y1, z1)=(x1 + x1 + 4, y1 + y1 + 4, z1 + z1 + 4) we find x1=x2=x3=-4 for c=5 you will find that c*(-4,-4,-4)=5*(-4,-4,-4)=(-4,-4,-4) if you calculate you will find "absurd". So R^3 with thos operation can not form a vectorspace.
 November 21st, 2011, 07:47 AM #3 Newbie   Joined: Nov 2011 Posts: 2 Thanks: 0 Re: Vector spaces Thanks!
 January 27th, 2014, 07:13 AM #4 Newbie   Joined: Jan 2014 Posts: 2 Thanks: 0 Re: Vector spaces Can you guys also help me with this problem, please? If these two operations are defined as follows. With these new definitions, is R2 a vector space? (x1, y1) + (x2, y2) = (x1, 0) and c(x, y) = (cx, cy) If the answer is no, can you list all axioms that fail.(I know there are 10 axioms) Thanks you so much for your help
 January 30th, 2014, 06:41 AM #5 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Vector spaces You know there are 10 axioms? Do you know what those axioms are? Can you not, yourself, check to see which of those axioms are true?
 January 30th, 2014, 01:39 PM #6 Senior Member   Joined: Mar 2012 Posts: 294 Thanks: 88 Re: Vector spaces Is there ANY vector (a,b) in R^2 that will serve as the 0-vector under that operation?

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### (x1,y1,z1) (x2,y2,z2)=(x1-x2, y1-y2, z1-z2)

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