1.1 Exercises

20 points · 4 extra credit

Systems of Linear Equations

1. 1. (2 pts) Solve the system by elimination or substitution:

   $\begin{aligned} x_1 + 2x_2 &= 4 \\ 3x_1 - x_2 &= 5. \end{aligned}$

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2. 2. (3 pts) Write the augmented matrix of the following system and use elementary row operations to solve it:

   $\begin{aligned} x_1 + x_2 + x_3 &= 6 \\ 2x_1 - x_2 + x_3 &= 3 \\ x_1 + 2x_2 - x_3 &= 2. \end{aligned}$

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3. 3. (3 pts) Consider the system

   $\begin{aligned} x_1 - 3x_2 &= 1 \\ 2x_1 + h\,x_2 &= 4. \end{aligned}$

   Find all values of $h$ for which the system has

   (a) a unique solution,

   (b) no solution,

   (c) infinitely many solutions.

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4. 4. (3 pts) Each matrix below is the augmented matrix of a linear system that has already been row reduced. In each case state whether the system is consistent, and if so how many solutions it has.

   (a) $\left[\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 2 \end{array}\right]$

   (b) $\left[\begin{array}{ccc|c} 1 & -2 & 0 & 4 \\ 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right]$

   (c) $\left[\begin{array}{ccc|c} 1 & 0 & 3 & 2 \\ 0 & 1 & -1 & 4 \\ 0 & 0 & 0 & 6 \end{array}\right]$

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) Every elementary row operation is reversible.

   (b) A system of two equations in three unknowns can have exactly one solution.

   (c) If a linear system is inconsistent, then the reduced form of its augmented matrix has a row of the form $[\,0 \;\; 0 \;\; \cdots \;\; 0 \mid b\,]$ with $b \neq 0$.

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6. 6. (3 pts) Explain why performing any one elementary row operation on the augmented matrix of a linear system does not change its solution set. Address all three types of operation (replacement, interchange, scaling).

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7. 7. (3 pts) Describe geometrically the solution set of each system in $\mathbb{R}^2$, and state how many solutions it has.

   (a) $\begin{aligned} x_1 - x_2 &= 1 \\ 2x_1 - 2x_2 &= 5 \end{aligned}$

   (b) $\begin{aligned} x_1 - x_2 &= 1 \\ 2x_1 - 2x_2 &= 2 \end{aligned}$

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8. 8. (4 pts ••) Suppose $u = (u_1, \dots, u_n)$ and $v = (v_1, \dots, v_n)$ are two distinct solutions of a system of linear equations. For a real number $t$, let $w$ be the tuple whose $k$-th entry is $w_k = u_k + t(u_k - v_k)$. Prove that $w$ is also a solution for every $t$, and conclude that the system has infinitely many solutions.

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