1.E: Exercises for Chapter 1

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Mar 5, 2021
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- 1: What is linear algebra
- 2: Introduction to Complex Numbers

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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 $x, y, z:$

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

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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