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I have trouble proving the following:

Suppose that $V$ is inner product space. Let $v_1,...,v_n$ be a basis of $V$, and let $w_1,...w_n$ be some vectors.

I need to show that,

$$G(w_1,...w_n) = C^TG(v_1,...v_n)C$$

Where $$G(v_1,...v_n)=\begin{pmatrix} <v_1,v_1> & <v_1,v_1> & &<v_1,v_n> \\ <v_2,v_1> & <v_2,v_2> & &<v_2,v_1> \\ ...& ... & ... & \\ <v_n,v_1>&... & ... & <v_n,v_n> \end{pmatrix}$$

And $C$ is matrix that in column $i$ have decomposition of $w_i$ with $v_i$'s

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The Matrix $G$ is called a gram matrix. You can begin with a inner product space of dimension 2 to understand the formula.

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  • $\begingroup$ Question: Let $A,B,C \in M_n(F)$. What is $(ABC)_{i,j}$? Is it - $(A)_i(B)_j(C)_{i}$? $\endgroup$ – 17SI.34SA May 6 '13 at 11:59

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