Let A 1 , A 2 , . . . , A r Some non-zero vectors in R n such that A i ⋅ A j = 0 for all i ≠ jLet c , . . . , c r some real numbers such that c 1 A 1 + c 2 A 2 + ⋯ + c r A r = 0Prove that all c i = 0First it is more simple to work with the special case r=2, then since A 1 ⋅ A 2 = 0 Then there is no such c 1 , c 2 ∈ R 2 such that c 1 A 1 + c 2 A 2 = 0except for the case where c 1 = c 2 = 0This last statement has to formalize in a more rigorous way, but i don’t how to do it i’ve just an intuition that this is the right case, and for higher dimensional vector this intuition would disappear.

Question

Let       A    1    ,      A    2    ,  .  .  .  ,      A    r   Some non-zero vectors in             R        n   such that       A    i    ⋅      A    j    =  0 for all   i  ≠  jLet   c  ,  .  .  .  ,      c    r   some real numbers such that      c    1        A    1    +      c    2        A    2    +  ⋯  +      c    r        A    r    =  0Prove that all       c    i    =  0First it is more simple to work with the special case r=2, then since       A    1    ⋅      A    2    =  0 Then there is no such       c    1    ,      c    2    ∈            R        2   such that      c    1        A    1    +      c    2        A    2    =  0except for the case where       c    1    =      c    2    =  0This last statement has to formalize in a more rigorous way, but i don’t how to do it i’ve just an intuition that this is the right case, and for higher dimensional vector this intuition would disappear.

Bradley Forbes · Accepted Answer

Use the properties of the dot product:                              c          1                          A          1                +                  c          2                          A          2                +        ⋯        +                  c          r                          A          r                =        0                                    ⇒                (                  c          1                          A          1                +                  c          2                          A          2                +        ⋯        +                  c          r                          A          r                )        ⋅                  A          1                =        0        ⋅                  A          1                                                                ⇒                          c          1                          A          1                ⋅                  A          1                +                  c          2                          A          2                ⋅                  A          1                +        ⋯        +                  c          r                          A          r                ⋅                  A          1                =        0                                                        ⇒                          c          1                =        0            Analogously, taking the dot product of the original equality with       A    2    ,  .  .  .  ,      A    r   we conclude that       c    2    =  0  ,  .  .  .  ,      c    r    =  0

polissemkt · Accepted Answer

c    i        |        |        A    i        |              |        2    =      A    i    ⋅      ∑          j      =      1        r        c    j        A    j    =      A    i    ⋅  0  =  0

Let A_1,A_2,...,A_r Some non-zero vectors in RR^n such that A_i * A_j=0 for all i != j. Let c,...,c_r some real numbers such that c_1A_1+c_2A_2+⋯+c_rA_r=0 Prove that all c_i=0

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