Re: [線代] orthonormal basis
※ 引述《mqazz1 (無法顯示)》之銘言：
: let <u,v> be an inner product defined on an n-dimensional real vector space V
: let S be an orthonormal basis of V
: if [u]_s = (u1, u2, ..., un)^T and [v]_s = (v1, v2, ..., vn)^T
: are the coordinates of u and v
: prove that <u,v> = u1v1 + u2v2 + ... + unvn
: http://ppt.cc/s9~Z
<sol>
By 題目, Let S={w_1,w_2,..,w_n} is an othonormal basis for V,
n n
u= Σ(u_i)*(w_i) v=Σ (v_j)*(w_j)
i=1 j=1
n n
<u,v>= <Σ(u_i)*(w_i) , Σ (v_j)*(w_j)>
i=1 j=1
n
= Σ (u_i)*(v_j)<w_i,w_j>
i=j=1
n
= Σ (u_i)(v_i) .
i=1
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