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I'm studying a proof and I have $j$ (the all-1 vector) represented in the basis of orthonormal vectors $\{v_1, ..., v_n\}$ such that

$$j = \sum_ic_i v_i$$

I don't understand why I then have:

$$j^Tv_i = c_i$$

and

$$\sum_ic_i^2 = n$$

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1 Answer 1

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Note that $$j^Tv_i=\left(\sum_i c_iv_i^T\right)v_i=c_i v_i^Tv_i$$ because $v_j^Tv_i=0$ if $j\ne i$. Hence we have $j^T v_i=c_i$. For the second equality note that $$n=j^Tj=\left(\sum_i c_iv_i^T\right)\left(\sum_i c_iv_i\right)$$ Now expand the product to get $\sum_i c_i^2 $ (use the fact that $v_i^Tv_j=0$ if $i\ne j$ and $1$ otherwise).

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