Systems of Linear Equations and Elementary Operations

Fundamentals of Linear Systems

• Linear Equation: An equation of the form $a_1x_1 + a_2x_2 + \dots + a_nx_n = b$, where $x_1, x_2, \dots, x_n$ are variables raised to the first power, $a_1, a_2, \dots, a_n$ are real coefficients, and $b$ is the constant term.

• System of Linear Equations: A finite collection of linear equations involving the same set of variables $x_1, x_2, \dots, x_n$.

• Solution: A sequence of numbers $s_1, s_2, \dots, s_n$ that satisfies every equation in the system when substituted for the corresponding variables.

• System Consistency:

  • Consistent System: A linear system that possesses at least one solution (either a unique solution or infinitely many solutions).

  • Inconsistent System: A linear system that has no solution.

• Parametric Form: A method of expressing an infinite set of solutions using independent variables called parameters (such as $s$ and $t$).

Geometric Interpretation in Two Variables

• Graphical Representation: In two variables, each linear equation $ax + by = c$ represents a straight line. The solutions to a system correspond to the points $P(s, t)$ common to all lines.

• Three Geometric Cases:

  • Unique Solution: The lines intersect at a single point.

• No Solution: The lines are parallel and distinct, meaning they never intersect.

• Infinitely Many Solutions: The lines are coincident (identical), sharing every point.

Matrix Representations

• Augmented Matrix: A rectangular array containing the coefficients of each variable followed by the constant terms, separated by a vertical dividing line.

• Coefficient Matrix: The matrix formed solely by the coefficients of the system's variables.

• Constant Matrix: The single-column matrix containing the constant terms from the right-hand side of each equation.

Elementary Operations and Row Operations

• Elementary Operations on Systems: Operations performed on linear equations to yield an equivalent system:

  • I. Interchange two equations.

  • II. Multiply one equation by a nonzero number.

  • III. Add a multiple of one equation to a different equation.

• Theorem 1.1.1: Performing a sequence of elementary operations on a system of linear equations yields an equivalent system with the identical set of solutions.

• Elementary Row Operations on Matrices: Operations performed directly on the rows of an augmented matrix:

  • I. Interchange two rows.

  • II. Multiply one row by a nonzero number.

  • III. Add a multiple of one row to a different row.

• Inverses of Elementary Operations: Every elementary operation can be reversed by an inverse operation of the same type:

  • Type I Inverse: Interchange the same two rows again.

  • Type II Inverse: Multiply the row by $\frac{1}{k}$ (where $k \neq 0$).

  • Type III Inverse: Subtract $k$ times row $p$ from row $q$ (or add $-k$ times row $p$ to row $q$).