SCC121 Fundamentals of Computer Science - Systems of Linear Equations

Flashcards covering linear equations, systems of equations, matrices, and the Gauss-Jordan elimination algorithm.

Linear Equation

An equation in which each variable is raised to the power of 1 and without product of variables.

Standard form of a linear equation in n variables

𝑎1 ∙ 𝑥1 + 𝑎2 ∙ 𝑥2 + ⋯ + 𝑎𝑛 ∙ 𝑥𝑛 = 𝑏

Slope-intercept form of a linear equation in 2 variables

𝑦 = 𝑚 ∙ 𝑥 + 𝑟, where 𝑚 is the slope and 𝑟 is the 𝑦-intercept

Solving a linear equation

Finding the set of values for each of its variables, for which the equation becomes a true statement.

System of linear equations

A finite set of two or more linear equations involving the same variables.

Solution set of a system of linear equations

The set of all ordered n-tuples of numbers that makes each equation a true statement when substituted for the variables.

Equivalent linear systems

Two linear systems with the same solution set.

Basic algebraic approach to solving systems of two linear equations

Simplify the problem by eliminating one of the variables to find the other through substitution, subtraction, or equating.

Geometric approach to solving systems of two linear equations

Representing the set of solutions as the intersection of two lines in ℝ2.

Consistent system

A system of linear equations that has either one solution or infinitely many solutions.

Inconsistent system

A system of linear equations that has no solution.

Geometric approach to solving systems of three linear equations

Representing the set of solutions as the intersection of three planes in ℝ3.

Basic algebraic approach to solving systems of linear equations

Simplify the problem by eliminating one of the variables to find the other through substituting, subtracting, or equating.

Matrix

A rectangular array that compactly records the essential information of a linear system.

Coefficient matrix

A matrix consisting of the coefficients of the variables in a system of linear equations.

Augmented matrix

The coefficient matrix with an added column containing the constants from the right side of the equations.

Gauss-Jordan elimination algorithm

An advanced systematic approach for solving systems of linear equations.

Leading entry (pivot)

The first non-zero entry in a row of a matrix, starting from the left.

Row echelon form

A matrix transformed by row operations to solve systems of linear equations.

Row-reduced echelon form (RREF)

A matrix satisfying specific conditions, including leading entries of 1, entries below leading entries being zero, and more.

Forward phase

In Gauss-Jordan elimination, move left to right to obtain pivot and create zeros, resulting in row echelon form.

Backward phase

In Gauss-Jordan elimination, move right to left for pivot, eliminate numbers above pivot, leading to reduced row echelon form.

Row operations in Gauss-Jordan Elimination Algorithm

Swap the position of two rows, multiply a row by a non-zero constant, and add a multiple of one row to another row.