Answer
See sxplanation
Work Step by Step
Given:
Linear System:
\[
\begin{array}{l}
4 x_{1}-2 x_{2}+7 x_{3}=-5 \\
8 x_{1}-3 x_{2}+10 x_{3}=-3
\end{array}
\]
Goal:
a.) Define appropriate vectors, re-write the system in terms of linear combinations. Determine if the
system is consistent.
b.) Define an appropriate matrix. Restate the problem using "columns of $\mathrm{A}^{\prime \prime}$
c.) Define an appropriate linear transformation, $T$, using the matrix created in $b .$ Restate the problem in terms of $T$
3
Concepts:
Linear Combination
Vector Column
Linear Transformation
\[
\text { put } \mathbf{a}_{1}=\left[\begin{array}{l}
4 \\
8
\end{array}\right], \mathbf{a}_{2}=\left[\begin{array}{l}
-2 \\
-3
\end{array}\right], \mathbf{a}_{3}=\left[\begin{array}{c}
7 \\
10
\end{array}\right], \mathbf{b}=\left[\begin{array}{l}
-5 \\
-3
\end{array}\right]
\]
For the given vectors, the system is represented by the following vector equation:
\[
x_{1}\left[\begin{array}{l}
4 \\
8
\end{array}\right]+x_{2}\left[\begin{array}{l}
-2 \\
-3
\end{array}\right]+x_{3}\left[\begin{array}{c}
7 \\
10
\end{array}\right]=\left[\begin{array}{c}
-5 \\
-3
\end{array}\right]
\] make an augmented matrix for the vector equation, then decrease the matrix to row echelon form to determine if $\mathbf{b}$ is a linear combination of $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$
\[
\left[\begin{array}{cccc}
4 & -2 & 7 & -5 \\
8 & -3 & 10 & -3
\end{array}\right] \sim\left[\begin{array}{cccc}
4 & -2 & 7 & 5 \\
0 & 0 & -4 & 7
\end{array}\right]
\]
Solve
The system in (a) is consistent. Therefore, b is a linear combination of $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$
\[
\text { Let } A=\left[\begin{array}{cccc}
4 & -2 & 7 & -5 \\
8 & -3 & 10 & -3
\end{array}\right]
\]
Determine if $\mathbf{b}$ is a linear combination of the columns
of $A$
Solve
$\operatorname{Let} T(\mathbf{x})=A \mathbf{x}$
