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bugatti79

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I am looking at my notes. Wondering where the highlighted comes from.

Prove that a finite orthogonal set is lineaarly independent

let u=(x_1,x_2,x_n) bee an orthogonal set set of vectors in an ips.

To show u is linearly independent suppose

Ʃ ##\alpha_i x_i=0## for i=1 to n

Fix any j=1 and consider <Ʃ##\alpha_i x_i, x_j##> i=1 to n

then

0=<Ʃ##\alpha_i x_i, x_j##> i=1 to n

=Ʃ<##\alpha_i x_i, x_j##> i=1 to n

=Ʃ##\alpha_i <x_i, x_j>## i=1 to n

=##\alpha_j <x_j, x_j>## since u is an orthonormal set

Where does this line come from? Thanks