If \(\mathbf{B}\in M_{m,n}(\mathbb{K})\) is the reduced row echelon form of \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) then \(\operatorname{Ker}(\mathbf{A})=\operatorname{Ker}(\mathbf{B}).\)

• True
• False

If \(\mathbf{B}\in M_{m,n}(\mathbb{K})\) is the reduced row echelon form of \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) then \(\operatorname{Im}(\mathbf{A})=\operatorname{Im}(\mathbf{B}).\)

• True
• False

If \(\mathbf{B}\in M_{n,n}(\mathbb{K})\) is the reduced row echelon form of the invertible matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K}),\) then \(\operatorname{Im}(\mathbf{A})=\operatorname{Im}(\mathbf{B}).\)

• True
• False

If \(\mathbf{B}\in M_{m,n}(\mathbb{K})\) is the reduced row echelon form of \(\mathbf{A}^T\in M_{n,m}(\mathbb{K}),\) then \(\operatorname{Im}(\mathbf{A})=\operatorname{Im}(\mathbf{B}^T).\)

• True
• False

If \(\mathbf{B}\in M_{m,n}(\mathbb{K})\) is the reduced row echelon form of \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) then \(\operatorname{rank}(\mathbf{A})=\operatorname{rank}(\mathbf{B}).\)

• True
• False

If \(\mathbf{B}_1,\ldots,\mathbf{B}_N,\mathbf{C}_1,\ldots,\mathbf{C}_{\tilde{N}}\in M_{m,n}(\mathbb{K})\) are such that \(\mathbf{B}_N\mathbf{B}_{N-1}\cdots \mathbf{B}_1 = \mathbf{C}_{\tilde{N}}\mathbf{C}_{\tilde{N}-1}\cdots \mathbf{C}_1\) is the reduced row echelon form of \(\mathbf{A}\in M_{m,n}(\mathbb{K}),\) then \(\tilde N=N\) and \(\mathbf{B}_k = \mathbf{C}_k\) for all \(k=1,\ldots, N.\)

• True
• False

\(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is invertible if and only if the reduced row echelon form of \(\mathbf{A}\) equals \(\mathbf{1}_{n}.\)

• True
• False

In general, a solution to the equation \(\mathbf{A}\vec x = \vec b\) is given by \(\vec x = \mathbf{A}^{-1}\vec b,\) for \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) and \(\vec x,\vec b\in\mathbb{K}^n.\)

• True
• False

If \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is invertible, the equation \(\mathbf{A}\vec x = \vec b,\) where \(\vec x,\vec b\in\mathbb{K}^n\) always has a unique solution.

• True
• False

If \(\mathbf{A}\in M_{m,n}(\mathbb{K})\) has full rank, the equation \(\mathbf{A}\vec x = \vec b,\) where \(\vec x\in\mathbb{K}^n,\vec b\in\mathbb{K}^m\) always has a at least a solution.

• True
• False

If \(\mathbf{A}\in M_{m,n}(\mathbb{K})\) is such that \(f_\mathbf{A}:\mathbb{K}^n\to\mathbb{K}^m\) is surjective, the space of solutions of the equation \(\mathbf{A}\vec x = \vec b\) forms a vector space of dimension \(n-m.\)

• True
• False

If \(\mathbf{A}\in M_{m,n}(\mathbb{K})\) is such that \(f_\mathbf{A}:\mathbb{K}^n\to\mathbb{K}^m\) is surjective, then the equation \(\mathbf{A}\vec x = \vec b\) never has a unique solution.

• True
• False

If a subset of \(\mathbb{K}^n\) spans \(\mathbb{K}^n,\) it is linearly independent.

• True
• False

If \(S\) is a linearly dependent subset of \(\mathbb{K}^n,\) so is every subset of \(\mathbb{K}^n\) that contains \(S.\)

• True
• False

Every subset of a linearly independent set is linearly independent.

• True
• False

Any linearly independent set of \(n\) vectors in \(\mathbb{K}^n\) is a basis for \(\mathbb{K}^n.\)

• True
• False