1: Linear systems

Section 1: Linear Systems

A linear equation in variables $x_1, x_2, ..., x_n$ has the form: $a_1x_1 + a_2x_2 + ... + a_nx_n = b$ where $a_1, a_2, ..., a_n$ (coefficients) and $b$ (constant term) are real or complex numbers. A system of linear equations is a collection of one or more such equations involving the same variables. For example:

$$\begin{cases} 2x_1 + 3x_2 = 8 & (1) \\ x_1 - x_2 = 1 & (2) \end{cases}$$

A solution to the system is an assignment of numbers to the variables $(s_1, s_2, ..., s_n)$ that satisfies every equation simultaneously. The solution set is the collection of all possible solutions. Systems are classified based on their solutions:

• Consistent: Has at least one solution.
  • Unique Solution: Exactly one solution (e.g., $x_1 = 2, x_2 = 1$ for the system above).
  • Infinitely Many Solutions: An infinite number of solutions (typically expressed parametrically).
• Inconsistent: Has no solution (e.g., parallel lines that never intersect).

Geometric Interpretation provides crucial intuition:

• In $\mathbb{R}^2$ (two variables): Each equation represents a line. The solution(s) correspond to the point(s) where the lines intersect (one point = unique solution, same line = infinite solutions, parallel lines = inconsistent).
• In $\mathbb{R}^3$ (three variables): Each equation represents a plane. Solutions correspond to points, lines, or planes of intersection (e.g., three planes intersecting at a single point, along a line, or having no common point).

A system is homogeneous if all constant terms $b$ are zero:

$$\begin{cases} a_{11}x_1 + ... + a_{1n}x_n = 0 \\ \vdots \\ a_{m1}x_1 + ... + a_{mn}x_n = 0 \end{cases}$$

Homogeneous systems are always consistent because the trivial solution $(x_1 = 0, x_2 = 0, ..., x_n = 0)$ always satisfies them. The key question is whether non-trivial solutions (infinitely many) exist. This depends on the relationship between the number of equations ($m$) and variables ($n$), and the linear dependence of the equations – concepts explored further via row reduction and vector spaces.