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a. Check all of the statements below that must be true if A is an $m \times n$ matrix with $n$ pivot columns. A. There exists an $n \times m$ matrix Y such that $YA = I$, where I denotes the $n \times n$ identity matrix. \checkmark B. The transformation $x \mapsto Ax$ is injective (one-to-one). \checkmark C. The columns of A are linearly independent. D. The transformation $x \mapsto Ax$ is surjective (onto). \checkmark E. For any positive integer $p$ and any $p \times m$ matrices X and Y, if $XA = YA$ then $X = Y$. b. Check all of the statements below that must be true if A is an $m \times n$ matrix with $m$ pivot columns. A. $A^T A$ is invertible. B. For any $b \in \mathbb{R}^m$, the equation $Ax = b$ has at least one solution. \checkmark C. For any $b \in \mathbb{R}^m$, the equation $Ax = b$ has at most one solution. \checkmark D. For any positive integer $p$ and any $p \times m$ matrices X and Y, if $XA = YA$ then $X = Y$. E. $AA^T$ is invertible.

          a. Check all of the statements below that must be true if A is an $m \times n$ matrix with $n$ pivot columns.
A. There exists an $n \times m$ matrix Y such that $YA = I$, where I denotes the $n \times n$ identity matrix.
\checkmark B. The transformation $x \mapsto Ax$ is injective (one-to-one).
\checkmark C. The columns of A are linearly independent.
D. The transformation $x \mapsto Ax$ is surjective (onto).
\checkmark E. For any positive integer $p$ and any $p \times m$ matrices X and Y, if $XA = YA$ then $X = Y$.
b. Check all of the statements below that must be true if A is an $m \times n$ matrix with $m$ pivot columns.
A. $A^T A$ is invertible.
B. For any $b \in \mathbb{R}^m$, the equation $Ax = b$ has at least one solution.
\checkmark C. For any $b \in \mathbb{R}^m$, the equation $Ax = b$ has at most one solution.
\checkmark D. For any positive integer $p$ and any $p \times m$ matrices X and Y, if $XA = YA$ then $X = Y$.
E. $AA^T$ is invertible.

a. Check all of the statements below that must be true if A is an m × n matrix with n pivot columns.
A. There exists an n × m matrix Y such that YA = I, where I denotes the n × n identity matrix.
B. The transformation x ↦ Ax is injective (one-to-one).
C. The columns of A are linearly independent.
D. The transformation x ↦ Ax is surjective (onto).
E. For any positive integer p and any p × m matrices X and Y, if XA = YA then X = Y.
b. Check all of the statements below that must be true if A is an m × n matrix with m pivot columns.
A. A^T A is invertible.
B. For any b ∈ℝ^m, the equation Ax = b has at least one solution.
C. For any b ∈ℝ^m, the equation Ax = b has at most one solution.
D. For any positive integer p and any p × m matrices X and Y, if XA = YA then X = Y.
E. AA^T is invertible.

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