1.E: Exercises for Chapter 1

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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.

Contributors

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