Let \(V\) be a \(\mathbb{K}\)-vector space and \(\mathcal S,\mathcal T\subset V.\) If \(\mathcal S\cup \mathcal T\) is a basis of \(V,\) then \(V=\operatorname{span}(\mathcal S)\oplus\operatorname{span}(\mathcal T).\)

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Let \(V\) be a \(\mathbb{K}\)-vector space and \(\mathcal S,\mathcal T\subset V.\) If \(\mathcal S\cup \mathcal T\) is a basis of \(V\) and \(\mathcal S\cap\mathcal T=\emptyset\) then \(V=\operatorname{span}(\mathcal S)\oplus\operatorname{span}(\mathcal T).\)

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• False

If \(U_1,U_2,U_3\subset V\) are subspaces of a \(\mathbb{K}\)-vector space, then \(U_1+U_2=U_1+U_3\) implies \(U_2=U_3.\)

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If \(U_1,U_2,U_3\subset V\) are subspaces of a \(\mathbb{K}\)-vector space, then \(U_1+U_2=U_1+U_3\) implies \(\dim(U_2)=\dim(U_3).\)

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If \(U_1,U_2,U_3\subset V\) are subspaces of a finite-dimensional \(\mathbb{K}\)-vector space, then \(U_1\oplus U_2=U_1\oplus U_3\) implies \(\dim(U_2)=\dim(U_3).\)

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If \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) then \(\operatorname{Tr}(\mathbf{A}^T\mathbf{A}) = \operatorname{Tr}(\mathbf{A}\mathbf{A}^T).\)

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If \(\mathbf{A},\mathbf{B}\in M_{m,n}(\mathbb{K}),\) then \(\operatorname{Tr}(\mathbf{A}+\mathbf{B})=\operatorname{Tr}(\mathbf{A})+\operatorname{Tr}(\mathbf{B}).\)

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• False

If \(\mathbf{A},\mathbf{B}\in M_{m,n}(\mathbb{K})\) and \(\mathbf{B}\) is invertible, then \(\operatorname{Tr}(\mathbf{B}\mathbf{A}\mathbf{B}^{-1})=\operatorname{Tr}(\mathbf{A}).\)

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If \(\mathbf{A},\mathbf{B}\in M_{m,n}(\mathbb{K})\) and \(\mathbf{B}\) is invertible, then \(\det(\mathbf{B}\mathbf{A}\mathbf{B}^{-1})=\det(\mathbf{A}).\)

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If \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) then \(\operatorname{Tr}(\mathbf{A})\ne\operatorname{Tr}(\mathbf{A}^T).\)

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If \(\mathbf{A},\mathbf{B},\mathbf{C}\in M_{n,n}(\mathbb{K}),\) then \(\operatorname{Tr}(\mathbf{A}\mathbf{B}\mathbf{C})=\operatorname{Tr}(\mathbf{B}\mathbf{C}\mathbf{A}).\)

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• False

If \(\mathbf{A},\mathbf{B},\mathbf{C}\in M_{n,n}(\mathbb{K}),\) then \(\operatorname{Tr}(\mathbf{A}\mathbf{B}\mathbf{C})=\operatorname{Tr}(\mathbf{B}\mathbf{A}\mathbf{C}).\)

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• False

If \(\mathbf{A},\mathbf{B},\mathbf{C}\in M_{n,n}(\mathbb{K}),\) then \(\det(\mathbf{A}\mathbf{B}\mathbf{C})=\det(\mathbf{B}\mathbf{A}\mathbf{C}).\)

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Let \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K}).\) If \(\operatorname{Tr}(\mathbf{A})=\operatorname{Tr}(\mathbf{B}),\) then \(\mathbf{A}\) and \(\mathbf{B}\) are conjugate.

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Let \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K}).\) If \(\det(\mathbf{A})=\det(\mathbf{B}),\) then \(\mathbf{A}\) and \(\mathbf{B}\) are conjugate.

• True
• False