1. Mark each statement as true or false and justify your answer.
(a) An example of a linear combination of vectors $\vec{v_1}$ and $\vec{v_2}$ is the vector $\frac{1}{2}\vec{v_1}$.
(b) When $\vec{v}$ and $\vec{u}$ are nonzero vectors, span{$\vec{v}$, $\vec{u}$} contains only the line through $\vec{u}$ and the
origin and the line through $\vec{v}$ and the origin.
(c) Asking whether the linear system corresponding to an augmented matrix
$\begin{bmatrix} \vec{a_1} & \vec{a_2} & \vec{a_3} & \vec{b} \end{bmatrix}$ has a solution amounts to asking whether $\vec{b}$ is in span{$\vec{a_1}$, $\vec{a_2}$, $\vec{a_3}$}.
(d) A vector $\vec{b}$ is a linear combination of the columns of a matrix A if and only if the equation
$A\vec{x} = \vec{b}$ has at least one solution.
(e) The equation $A\vec{x} = \vec{b}$ is consistent if the augmented matrix $[A \vec{b}]$ has a pivot position
in every row.
(f) If the columns of an $m \times n$ matrix A span $\mathbb{R}^m$, then the equation $A\vec{x} = \vec{b}$ is consistent
for each $\vec{b} \in \mathbb{R}^m$.
(g) If the coefficient matrix A has a pivot position in every row, then the equation $A\vec{x} = \vec{b}$
is inconsistent.
(h) If A is an $m \times n$ matrix and if the equation $A\vec{x} = \vec{b}$ is inconsistent for some $\vec{b} \in \mathbb{R}^m$,
then A cannot have a pivot position in every row.
(i) If the columns of an $m \times n$ matrix A span $\mathbb{R}^m$, then the equation $A\vec{x} = \vec{b}$ is consistent
for each $\vec{b} \in \mathbb{R}^m$.
(j) A homogeneous system of equations can be inconsistent.
