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please, show me Let a, b R3. Suppose that for ever.pdf

26 Mar 2023
 please, show me Let a, b  R3. Suppose that for ever.pdf
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please, show me Let a, b R3. Suppose that for ever.pdf

  1. please, show me Let a, b R3. Suppose that for every v R3, v middot a = v middot b. Prove that a = b. Hint: Write a = (a1, a2, a3) and b = (b1, b2, b3). Make some clever choices of v and use the hypothesis of this problem to show that a1 = b1, a2 = b2 and a3 = b3 (which means a = b). Solution since v.a=v.b is true for every v in R^3, we can take any v. take v=(1,0,0) in R^3 then v.a=(1,0,0).(a1,a2,a3)=1.a1+0+0=a1 and v.b=(1,0,0).(b1,b2,b3)=1.b1+0+0=b1, but v.a=v.b implies a1=b1, similary take v=(0,1,0) we will get a2=b2, take v=(0,0,1) we will get a3=b3. hence a=b
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