SOLUTION: Given V=R^2 with "non-standard" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real number),
u(+)v=[(x1^1/3 + x^2^1/3)^3, (y1^1/3 + y2^1/3)^3] and
c(*)v=(c^3*x2, c^3*y2), w
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Question 32261: Given V=R^2 with "non-standard" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real number),
u(+)v=[(x1^1/3 + x^2^1/3)^3, (y1^1/3 + y2^1/3)^3] and
c(*)v=(c^3*x2, c^3*y2), where (*)and(+)anr non-standard operations,
I want to prove that V form a vector space, using 10 vector space axioms. But, I can't figure out how to handle the exponents (raised to the third power) in the operations...Also, I am confused about the "non-standard"...Can I still use some real number for x and y (like x=1, y=1)and 'c' when varifying the closure of the axioms? How exactly are "standard" and "non-standard" different when using the ten vector space axioms? For 'distributive property', could I use another vector w=(x3,y3)?
Would you help me with this problem???
Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
I AM ANSWERING YOUR DOUBTS AND LEAVING THE SOLUTION TO YOU.IF YOU HAVE STILL DIFFICULTY PLEASE COME BACK AND WE SHALL SOVE IT FOR YOU...OK?
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Given V=R^2 with "non-standard" operations: for u=(x1,y1) and v=(x2,y2)in R^2, and c(real number),
u(+)v=[(x1^1/3 + x^2^1/3)^3, (y1^1/3 + y2^1/3)^3] and
c(*)v=(c^3*x2, c^3*y2), where (*)and(+)anr non-standard operations,
I want to prove that V form a vector space, using 10 vector space axioms.
GOOD.....PROCEED..THOUGH I AM NOT SURE ABOUT YOUR COUNT OF 10 AXIOMS..ANY WAY I HOPE THEY INCLUDE ALL REQUIRED STIPULATIONS.
But, I can't figure out how to handle the exponents (raised to the third power) in the operations...
SAME WAY AS YOU DO IN NORMAL ALGEBRA..THERE IS NO DIFFERENCE.THOUGH U AND V ARE VECTORS WITH SPECIAL PROPERTIES AS DEFINED,INDIVIDUALLY....C,X1,Y1 ETC...YOU CAN TREAT AS IN NORMAL MANNER.
Also, I am confused about the "non-standard"...
THIS WORD IS USED TO TELL THAT (+) IS NOT NORMAL + OR ADDITION...AS WE UNDERSTAND AND SIMILARLY (*) IS NOT NORMAL * OR MULTIPLICATION AS WE UNDERSTAND.YOU HAVE TO USE THE GIVEN FORMULAE TO DO THESE OPERATIONS OF (+) AND (*).BUT THE FORMULAE GIVEN FOR THOSE SPECIAL OR NON STANDARD OPERATION CONNECTING C,X1,Y1 ETC..HAS ^,+AND * WHICH ARE NORMAL OPERATIONS OF EXPONENTIATION,ADDITION AND MULTIPLICATION W.R.T THOSE VARIABLES.
Can I still use some real number for x and y (like x=1, y=1)and 'c' when varifying the closure of the axioms?
SURE ..THAT IS WHAT I HAD WRITTEN ABOVE.
How exactly are "standard" and "non-standard" different when using the ten vector space axioms?
I HOPE I EXPLAINED THIS IN DETAIL ABOVE..IF YOU ARE DEALING WITH U,V,OR W...THE VECTORS ...YOU HAVE TO USE THE GIVEN FORMULAE FOR * OR + AS SHOWN DISTINCTLY BY (*),(+)......BUT WHEN YOU ARE DEALING WITH C,X1,Y1,ETC...INDIVIDUALLY,YOU CAN DO NORMAL ADDITION MULTIPLICATION ETC...
For 'distributive property', could I use another vector w=(x3,y3)?
SURE..YOU SIMPLY WRITE LET W (X3,Y3) BE A VECTOR AND PROCEED...
Would you help me with this problem???
I TRUST I DID THAT TO YOUR SATISFACTION...GOOD LUCK!IF YOU NEED MORE PLEASE COME BACK
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