Let u 1 = ( a 1 , b 1 , c 1 , d 1 ) and u 2 = ( a 2 , b 2 , c 2 , d 2 ) be any vectors in R 4 Which inner product axioms do not hold with the definition ⟨ u 1 , u 2 ⟩ = a 1 a 2 + 2 b 1 b 2 − c 1 c 2 + 2 d 1 d 2 Now I believe that symmetry and positivity would hold. It seems that homogeneity could hold, as the scalar doesn't affect the order of multiplication. This would mean additivity is the only one that does not hold. If that is correct, I am having trouble constructing it to show it fails. Cause wouldn't ( a 1 + a 2 ) w 1 + 2 ( b 1 + b 2 ) w 2 − ( c 1 + c 2 ) w 3 + 2 ( d 1 + d 2 ) w 4 still be an instance of ⟨ u 1 , w ⟩ + ⟨ u 2 , w ⟩ ?

Question

Let       u    1    =  (      a    1    ,      b    1    ,      c    1    ,      d    1    ) and       u    2    =  (      a    2    ,      b    2    ,      c    2    ,      d    2    ) be any vectors in             R        4   Which inner product axioms do not hold with the definition      ⟨          u      1        ,          u      2        ⟩    =      a    1        a    2    +  2      b    1        b    2    −      c    1        c    2    +  2      d    1        d    2  Now I believe that symmetry and positivity would hold. It seems that homogeneity could hold, as the scalar doesn&#039;t affect the order of multiplication. This would mean additivity is the only one that does not hold. If that is correct, I am having trouble constructing it to show it fails. Cause wouldn&#039;t   (      a    1    +      a    2    )      w    1    +  2  (      b    1    +      b    2    )      w    2    −  (      c    1    +      c    2    )      w    3    +  2  (      d    1    +      d    2    )      w    4   still be an instance of       ⟨          u      1        ,    w    ⟩    +      ⟨          u      2        ,    w    ⟩  ?

Geovanni Shelton · Accepted Answer

It is not an inner product because the inner product of   (  0  ,  0  ,  1  ,  0  ) with itself is   −  1  &amp;lt;  0.   ⟨  u  ,  u  ⟩  ≥  0 is a necessary condition for   ⟨  .  ,  .  ⟩ to be an inner product.[Linearity in each variable is true in this case].

Let u_1=(a_1,b_1,c_1,d_1) and u_2=(a_2,b_2,c_2,d_2) be any vectors in RR^4 Which inner product axioms do not hold with the definition ⟨u_1,u_2⟩=a_1 a_2+2b_1 b_2−c_1 c_2+2 d_1 d_2

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