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

Yair Valentine

Yair Valentine

Answered question

2022-08-11

Let A 1 , A 2 , . . . , A r Some non-zero vectors in R n such that A i A j = 0 for all i j
Let c , . . . , c r some real numbers such that
c 1 A 1 + c 2 A 2 + + c r A r = 0
Prove that all c i = 0
First 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 = 0
except for the case where c 1 = c 2 = 0
This 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.

Answer & Explanation

Bradley Forbes

Bradley Forbes

Beginner2022-08-12Added 12 answers

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

polissemkt

Beginner2022-08-13Added 3 answers

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

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