Let \(\mathbf{b}=(v_1,v_2,v_3)\) be an ordered basis of a vector space \(V.\) Then \(\mathbf{c}= (2v_1+v_2,v_2+3v_3,v_1-v_3)\) is an ordered basis and \[\mathbf{C}(\mathbf{c},\mathbf{b}) = \begin{pmatrix} 2 & 0 & 1\\ 1 & 1 & 0\\ 0 & 3 & -1 \end{pmatrix}.\]

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Let \(\mathbf{b}=(v_1,v_2,v_3)\) be an ordered basis of a vector space \(V.\) Then \(\mathbf{c}= (2v_1+v_2,v_2+v_3,2v_1+2v_2+v_3)\) is an ordered basis and \[\mathbf{C}(\mathbf{c},\mathbf{b}) = \begin{pmatrix} 2 & 0 & 2\\ 1 & 1 & 2\\ 0 & 1 & 1 \end{pmatrix}.\]

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Let \(\mathbf{b}=(\vec v_1,\ldots, \vec v_n)\) be an ordered basis of \(\mathbb{K}^n\) with corresponding linear coordinate system \(\boldsymbol{\beta}=f_\mathbf A.\) If the rows of \(\mathbf A\) are given by \(\vec \nu_1,\ldots ,\vec\nu_n,\) then \(\vec\nu_i\vec v_j = \delta_{ij}.\)

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• False

Let \(\mathbf{b}\) and \(\mathbf{c}\) be two ordered bases of \(\mathbb{K}^n,\) then \(\mathbf{C}(\mathbf{b},\mathbf{c})\) is always invertible.

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Let \(\mathbf{b}=(\vec v_1,\ldots, \vec v_n)\) be an ordered basis of \(\mathbb{K}^n\) and let \(\mathbf{e}\) be the standard ordered basis of \(\mathbb{K}^n.\) If \(\mathbf{B}\in M_{n,n}(\mathbb{K})\) is the matrix whose \(i\)-th column is given by \(\vec v_i,\) \(1\leqslant i \leqslant n,\) then \(\mathbf{C}(\mathbf{b},\mathbf{e}) = \mathbf{B}.\)

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Let \(\mathbf{b}=(\vec v_1,\ldots, \vec v_n)\) be an ordered basis of \(\mathbb{K}^n\) and let \(\mathbf{B}\in M_{n,n}(\mathbb{K})\) be invertible. Then \(\mathbf{c}=(\mathbf{B}\vec v_1,\ldots,\mathbf{B}\vec v_n)\) is another ordered basis of \(\mathbb{K}^n.\)

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Let \(\mathbf{b}=(\vec v_1,\ldots, \vec v_n)\) and \(\mathbf{c}=(\mathbf{B}\vec v_1,\ldots,\mathbf{B}\vec v_n)\) be ordered bases of \(\mathbb{K}^n,\) where \(\mathbf{B}\in M_{n,n}(\mathbb{K}).\) Then \(\mathbf{C}(\mathbf{b},\mathbf{c})=\mathbf{B}.\)

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• False

Let \(\mathbf{A}\in M_{3,2}(\mathbb{R}).\) Then \(f_\mathbf{A}\) admits a left inverse.

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• False

Let \(\mathbf{A}\in M_{3,2}(\mathbb{R})\) have maximal rank. Then \(f_\mathbf{A}\) admits a left inverse.

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Let \(\mathbf{A}\in M_{3,2}(\mathbb{R}).\) Then \(f_\mathbf{A}\) cannot admit a right inverse.

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• False

There is an invertible matrix \(\mathbf{A}\) such that \(\mathbf{A}\mathbf{A}^T = \mathbf{0}_n.\)

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• False

There is an invertible matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) such that \(\mathbf{A}\mathbf{A}^T = \mathbf{1}_{n}.\)

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• False

If \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K})\) are invertible, then so is \(\mathbf{A}\mathbf{B}.\)

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If \(\mathbf{A},\mathbf{B}\in M_{n,n}(\mathbb{K})\) are invertible, then so is \(\mathbf{A}+\mathbf{B}.\)

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

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• False