If \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}:\mathbb{R}^2\times \mathbb{R}^2\to \mathbb{R}\) is a bilinear form, where \(\mathbf{A}\in M_{2,2}(\mathbb{R}),\) then there exists a symmetric matrix \(\mathbf{B}\in M_{2,2}(\mathbb{R})\) such that \(\langle \vec x,\vec x\rangle_{\mathbf{A}}=\vec x^T\mathbf{B}\vec x\) for all \(\vec x\in\mathbb{R}^2.\)

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If \(V\) is a finite-dimensional real vector space and \(\langle\cdot{,}\cdot\rangle:V\times V\to\mathbb{R}\) is a symmetric bilinear form, then for any subspace \(U\subset V\) it holds that \(U\oplus U^{\perp}=V.\)

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Let \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) be symmetric and let \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}:\mathbb{K}^n\times\mathbb{K}^n\to\mathbb{K}.\) Then \(\operatorname{Ker}(\mathbf{A})^{\perp}=\mathbb{K}^n.\)

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Consider a vector space \(V\) equipped with a symmetric bilinear form \(\langle\cdot{,}\cdot\rangle.\) Then the subset of \(V\) defined by \(\{v\in V| \langle v,u\rangle =0, \ \forall \ u\in V \}\) is not a subspace of \(V.\)

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For every ordered basis \(\mathbf{b}\) of \(\mathbb{R}^n,\) there is a symmetric matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) such that \(\mathbf{b}\) is an orthogonal basis of \(\mathbb{R}^n\) with respect to \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}.\)

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For every ordered basis \(\mathbf{b}\) of \(\mathbb{R}^n,\) there is a unique symmetric matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) such that \(\mathbf{b}\) is an orthogonal basis of \(\mathbb{R}^n\) with respect to \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}.\)

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For every ordered basis \(\mathbf{b}\) of \(\mathbb{R}^n,\) there is a unique symmetric matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) such that \(\mathbf{b}\) is an orthonormal basis of \(\mathbb{R}^n\) with respect to \(\langle\cdot{,}\cdot\rangle_{\mathbf{A}}.\)

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For a given symmetric bilinear form \(\langle\cdot{,}\cdot\rangle\) on a vector space \(V,\) there are infinitely many ordered bases \(\mathbf{b}\) of \(V\) such that \(\mathbf{b}\) is orthogonal with respect to \(\langle\cdot{,}\cdot\rangle.\)

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For a given signature, there is a unique symmetric matrix \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) with that signature.

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All symmetric matrices with the same signature as \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) are given by \(\mathbf{C}^T\mathbf{A}\mathbf{C},\) where \(\mathbf{C}\in \mathrm{GL} (n,\mathbb{R}).\)

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• False