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Let B = $\{v_1, \dots, v_n\}$ be an orthonormal basis for Rn Let p = $[v_1, \dots, v_n$]. Prove that for any x, we have that the B-matrix of x is equal to the tranpose of P times x.

I am unsure as to how to begin this proof.

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I am not sure what you call "B-matrix of x", but I will assume that you are looking for the components of $x$ in the basis $B$. In other words, you are looking for numbers $b_i$ such that $$x = b_1\nu_1+\cdots+b_n\nu_n.$$

Since $B$ is orthonormal, for any $i$ and $j$, we have $$\nu_i^T\nu_j=\delta_{ij}.$$

Therefore,when you multiply $\nu^T_i$ by $x$ only $i$-th term in the sum survives and we get $$\nu_i^Tx=b_i$$

or

$$P^Tx=\{b_1,\dots,b_n\}^T.$$

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