- #1

TranscendArcu

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## Homework Statement

## The Attempt at a Solution

Since A is a vector in V and since the [itex]A_i[/itex] form a basis, we can write A as a linear combination of the [itex]A_i[/itex]. We write [itex]A = x_1 A_1 + ... + x_n A_n[/itex]. Thus, we have,[itex]<x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i>[/itex]. Because two orthogonal vectors, when multiplied via inner product together give the zero vector, we simplify,

[itex]0 = x_i <A_i,A_i>[/itex]. Because we have presumed that the [itex]A_i ≠ 0[/itex], we cannot have [itex]<A_i,A_i> = 0[/itex], so we must have that the [itex]x_i = 0[/itex]. So we have [itex]<A,A_i> = <x_1 A_1 + ... + x_n A_n,A_i> = 0 = x_1 <A_1,A_i> + ... + x_n <A_n,A_i> = x_i <A_i,A_i> = 0 <A_i,A_i> = <0,A_i> = 0. Which shows that A = 0.

I don't know if I've done this correctly. I feel like it's not entirely convincing towards the end. Advice?