<aside> 💡 The above relation requires knowing the relation $(AB)^T=B^TA^T$

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• this states that the value of the inner product of two vectors is independent of the reference frame in which they are represented
• alternate interpretation is that the reference frame transformations preserve both lengths of vectors and angles between them
• a matrix can be inserted between 2 vectors and will lead to the relation:

$$ \mathbf{x}F^T M_F \mathbf{y}F = \mathbf{x}F^T D{GF}^T D{GF} M_F D{GF}^T D_{GF}\mathbf{y}_F = \mathbf{x}_G^T M_G \mathbf{y}_G $$

$$ M_G \equiv D_{GF}M_FD_{GF}^T $$

<aside> 💡 The above relation requires knowing the relation $(AB)^T=B^TA^T$

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<aside> 📢 The definition for $M_ G$ defines how matrices must transform under reference frame transformations otherwise the above relation will not hold.

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