Question

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.

          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.

1. Mark each statement as true or false and justify your answer.
(a) An example of a linear combination of vectors v⃗1⃗ and v⃗2⃗ is the vector (1)/(2)v⃗1⃗.
(b) When v⃗ and u⃗ are nonzero vectors, spanv⃗, u⃗ contains only the line through u⃗ and the
origin and the line through v⃗ and the origin.
(c) Asking whether the linear system corresponding to an augmented matrix
< b m a t r i x > has a solution amounts to asking whether b⃗ is in spana⃗1⃗, a⃗2⃗, a⃗3⃗.
(d) A vector b⃗ is a linear combination of the columns of a matrix A if and only if the equation
Ax⃗ = b⃗ has at least one solution.
(e) The equation Ax⃗ = b⃗ is consistent if the augmented matrix [A b⃗] has a pivot position
in every row.
(f) If the columns of an m × n matrix A span ℝ^m, then the equation Ax⃗ = b⃗ is consistent
for each b⃗∈ℝ^m.
(g) If the coefficient matrix A has a pivot position in every row, then the equation Ax⃗ = b⃗
is inconsistent.
(h) If A is an m × n matrix and if the equation Ax⃗ = b⃗ is inconsistent for some b⃗∈ℝ^m,
then A cannot have a pivot position in every row.
(i) If the columns of an m × n matrix A span ℝ^m, then the equation Ax⃗ = b⃗ is consistent
for each b⃗∈ℝ^m.
(j) A homogeneous system of equations can be inconsistent.

Added by Deanna A.

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