Chapter 1: Linear Equations in Linear Algebra

In Exercises 9 and 10, write the system first as a vector equation and then as a matrix equation.

10.$8x_1-x_2=4$

$\begin{array}{c}5x_1+4x_2=1\\ x_1-3x_2=2\end{array}$

In Exercises 10, write a vector equation that is equivalent tothe given system of equations.

10. $4x_1+x_2+3x_3=9$

$\begin{array}{c}{x}_1-7x_2-2x_3=2\\ 8x_1+6x_2-5x_3=15\end{array}$

Find the general solutions of the systems whose augmented matrices are given in Exercises 10.

10. $\left[\begin{array}{cccc}1& -2& -1& 3\\ 3& -6& -2& 2\end{array}\right]$

Balance the chemical equations in Exercise 5-10 using the vector equation approach discussed in this section.

The chemical reaction below can be used in some industrial processes, production of arsene$\left(As{H}_3\right)$. Use exact arithmetic or rational format for calculations to balance this equation.

$MnS+A{s}_{2}C{r}_{10}{O}_{35}+H_{2}S{O}_4\to HMn{O}_4+As{H}_3+Cr{S}_3{O}_{12}+H_{2}O$

In a certain region, about 6% of city’s population moves to the surrounding suburbs each year, and about 4% of the suburban population moves into the city. In 2015, there were, 10,000,000 residents in the city and 800,000 in the suburbs. Set up a difference equation that describes this situation, where${\mathbf{x}}_{\mathbf{0}}$is the initial population in 2015. Then estimate the population in the city and in the suburbs two years later, in 2017.

For Exercises 9 and 10, find all x in ${\mathbb{R}}^{\mathbf{4}}$ that are mapped into the zero vector by the transformation $\mathbf{x}|\to A\mathbf{x}$ for the given matrix A.

10. $A=\left[\begin{array}{cccc}1& 3& 9& 2\\ 1& 0& 3& -4\\ 0& 1& 2& 3\\ -2& 3& 0& 5\end{array}\right]$

In Exercises 7–10, the augmented matrix of a linear system has been reduced by row operations to the form shown. In each case, continue the appropriate row operations and describe the solution set of the original system.

10. $\left(\begin{array}{rlrlr}\ast 20c1& -2& 0& 3& -2\\ 0& 1& 0& -4& 7\\ 0& 0& 1& 0& 6\\ 0& 0& 0& 1& -3\end{array}\right)$

In Exercise 1-10, assume that $T$ is a linear transformation. Find the standard matrix of $T$.

$T:{\mathbb{R}}^2\to {\mathbb{R}}^2$, first reflects points through the vertical $x_2$-axis and then rotates points $\frac{\pi }{2}$ radians.

In Exercises 7-12, describe all solutions of $Ax=0$ in parametric vector form, where $A$ is row equivalent to the given matrix.

10. $\left[\begin{array}{cccc}1& 3& 0& -4\\ 2& 6& 0& -8\end{array}\right]$

In Exercises 9 and 10, (a) for what values of h is ${\mathbf{v}}_3$ inSpan $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\}$, and (b) for what values of $h$ is $\left\{{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3\right\}$ linearlydependent? Justify each answer.

10. ${\mathbf{v}}_1=\left[\begin{array}{c}1\\ -5\\ -3\end{array}\right]$, ${\mathbf{v}}_2=\left[\begin{array}{c}-2\\ 10\\ 6\end{array}\right]$, ${\mathbf{v}}_3=\left[\begin{array}{c}2\\ -9\\ h\end{array}\right]$