A linear map \(f:\mathbb{K}^n\to\mathbb{K}^m\) cannot be surjective if \(n<m.\)

• True
• False

A linear map \(f:\mathbb{K}^n\to\mathbb{K}^m\) cannot be injective if \(n>m.\)

• True
• False

If \(\mathcal S\) is a generating set of \(V\) and \(f:V\to W\) is an injective linear map, then \(f(\mathcal S)\) is a basis of \(W.\)

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• False

If \(\mathcal S\) is a generating set of \(V\) and \(f:V\to W\) is an injective linear map, then \(f(\mathcal S)\) is a generating set of \(W.\)

• True
• False

If \(\mathcal S,\mathcal T\) are generating sets of \(V\) with finitely many elements, then \(\operatorname{Card}(\mathcal S) = \operatorname{Card}(\mathcal T).\)

• True
• False

If \(\mathcal S,\mathcal T\) are bases of \(V\) with finitely many elements, then \(\operatorname{Card}(\mathcal S) = \operatorname{Card}(\mathcal T).\)

• True
• False

A set of vectors \({v_1,\dots,v_n}\in V\) is linearly independent if its span is \(n\)-dimensional.

• True
• False

The dimension of a subspace \(U\) of \(\mathbb{K}^n\) is equal to the number of vectors in a basis for it.

• True
• False

The kernel space of an \(m \times n\) matrix is contained in \(\mathbb{K}^m.\)

• True
• False

The image of an \(m \times n\) matrix is contained in \(\mathbb{K}^n.\)

• True
• False

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

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• False

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

• True
• False

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

• True
• False

Let \(V\) be a finite-dimensional \(\mathbb{K}\)-vector space. If \(U,W\subset V\) are subspaces such that \(\operatorname{dim}(U)<\operatorname{dim}(W),\) then \(U\subset W.\)

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• False

Let \(V\) be a finite-dimensional \(\mathbb{K}\)-vector space. If \(U,W\subset V\) are subspaces such that \(U\subset W,\) then \(\operatorname{dim}(U)\leqslant \operatorname{dim}(W).\)

• True
• False