
# 1.E: Exercises for Chapter 1

#### Calculational Exercises

1. Solve the following systems of linear equations and characterize their solution sets.(I.e., determine whether there is a unique solution, no solution, etc.) Also, write each system of linear equations as a single function $$f : \mathbb{R}^n \rightarrow \mathbb{R}^m$$ for appropriate choices of $$m, n \in \mathbb{Z}_+ .$$
(a) System of 3 equations in the unknowns $$x, y, z, w:$$

$x + 2y − 2z + 3w = 2 \\ 0.2x + 4y − 3z + 4w = 5 \\ 5x + 10y − 8z + 11w = 12$

(b) System of 4 equations in the unknowns $$x, y, z:$$

$x + 2y − 3z \\ x + 3y + z \\ 2x + 5y − 4z \\ 2x + 6y + 2z$

(c) System of 3 equations in the unknowns $$x, y, z:$$

2. Find all pairs of real numbers $$x_1$$ and $$x_2$$ that satisfy the system of equations

$x_1 + 3x_2 = 2, \;\;\; \tag{1.12}$

$x_1 − x_2 = 1. \;\;\; \tag{1.13}$

#### Proof-Writing Exercises

1. Let $$a, b, c,$$ and $$d$$ be real numbers, and consider the system of equations given by

$ax_1 + bx_2 = 0,\;\;\; \tag{1.14}$

$cx_1 + dx_2 = 0 \;\;\; \tag{1.15}$

Note that $$x_1 = x_2 = 0$$ is a solution for any choice of $$a, b, c,$$ and $$d.$$ Prove that if $$ad − bc = 0$$, then $$x_1 = x_2 = 0$$ is the only solution.

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