MAT1503 Linear Algebra With Complete Solutions Download To Score Distinction

MAT1503 Linear Algebra With Complete Solutions Download To Score Distinction homogeneous linear equations - x₁ − 2x₂ − 3x₃ + x₄ = 0 solution of a linear system - The element is a solution of each equation soluti on set (general solution) - All solutions of a linear system with the number sequence as the elements ordered n -tuple - A linear solution written as (a ₁, a₂, ... , aⱼ) ordered pair - ordered n -tuple if n = 2 ordered triple - ordered n -tuple if n = 3 consistent system - A linear system that has at least one solution equivalent systems - Two systems of equations that have the same solution set inconsistent system - A linear system that has no solutions parameter - An assigned arbitrary value where the linear system has infinite solutions parametric equations - The solution expressed by the equations using parameters algebraic operations - 1) Add a multiple of one equation to another 2) Multiply an equa tion by a nonzero constant 3) Interchange two equations Elementary Row Operations - 1) Add a multiple of one row to another row 2) Multiply any row by a nonzero constant 3) Interchange two rows augmented matrix - An abbreviation of a linear system in a r ectangular array of numbers elementary matrix - A matrix that was (or could be) produced by performing a single Elementary Row Operation on an identity matrix identity matrix - A square matrix with 1's on the main diagonal and zeros everywhere else. Note A×I = A and I×A = A Row Echelon Form - A matrix that has leading ones on the main diagonal and zeros below the leading ones. Reduced Row Echelon Form - A matrix that has leading ones on the main diagonal and zeros above and below the leading ones. matrix - A rectangular array of numbers linear equation in n (j) unknowns - a₁x₁ + a₂x₂ + ... + aⱼxⱼ = b linear equation - x + 3y = 7 or x ₁ − 2x₂ − 3x₃ + x₄ = -1 (no products or roots of variables) system of linear equations (linear system) - A finite set of linear equations solution of a lin ear equation - A sequence of numbers for which the substitution with variables will make the equation a true statement leading variables - The variables corresponding to the leading 1's in the augmented matrix free variables - The variables that can be assigned an arbitrary value Gaussian Elimination - 1) Put the matrix in augmented matrix form 2) Use row operations to put the matrix in echelon form 3) Write the equations from the echelon form matrix 4) Solve the equations. Gauss -Jordan Elimination - 1) Put the matrix in augmented matrix form 2) Use row operations to put the matrix in reduced ec helon form 3) Write the equations from the echelon form matrix 4) Solve the equations. trivial solution - The solutions of the homogeneous linear systems are 0 non-trivial solution - The solutions of the homogeneous linear systems are infinite (free vari ables are used) Free Variable Theorem for Homogeneous Systems - If a homogeneous linear system has n unknowns, and its augmented matrix has r nonzero rows in reduced row echelon form, then the system has n - r free variables entries - The numbers in the matrix