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I'm trying to prove orthogonality in the following case:
$Q:\mathbb{R}^2\to\mathbb{R}^2$ is a linear transformation that maps from an orthonormal basis $\mathcal{B}=\{\vec{b}_1,\vec{b}_2\}$ of $\mathbb{R}^2$
to an Orthonormal basis then is Q orthogonal

How do I approach this?

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You can do this with dot products and such, but here's a geometric interpretation. If $Q$ sends an orthonormal basis to another one (in $\mathbb{R}^2$), then $Q$ has to be either a rotation matrix, or it rotates and reflects. Then the result immediately follows since any $2\times 2$ matrix with those properties it orthogonal.

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