Exam 1: Systems of Linear Equations

A linear system with infinitely many solutions must have more variables than equations.

Determine if the system is diagonally dominant. If not, then if possible rewrite the system so that it is diagonally dominant.

The values for the first few Gauss-Seidel iterations for a linear system are given. Find the values for the next iteration.

If a linear system has infinitely many solutions, then there are more variables than equations.

Use partial pivoting with Gaussian elimination to find the solutions to the system.

The points (-6, 0, -1), (3, 2, 0), and (0, 3, -1) lie on a plane ax + by + cz = 1. Use a system of linear equations to find the equation of this plane.

Use a system of linear equations to find the values , , and for the partial fraction decomposition

Compute the first four Gauss-Seidel iterations for the system in question 11, with the initial value of each variable set equal to . Then rewrite the system so that it is diagonally dominant, set the value of each variable to 0, and again compute four Gauss-Seidel iterations.

Convert the given system to an augmented matrix, and find all solutions by transforming to reduced echelon form and using back substitution.

Reorder the equations to put the following system of three equations with four unknowns in echelon form: ​

Convert the given system to an augmented matrix, and find all solutions by reducing to echelon form and using back substitution.

Find all solutions to the system

The values for the first few Jacobi iterations for a linear system are given. Find the values for the next iteration.

Use a system of linear equations to find the values of the coefficients a, b, c if with , , and .

Use a system of linear equations to find the equation of the parabola which passes through the points (1, 2), (2, 0), and (3, -4).

Suppose a system of equations has two free variables. How many leading variables are there?

If a linear system has no free variables, then there exists at most one solution.

Compute the first three Jacobi iterations, using as the initial value for each variable. Then find the exact solution and compare.

Convert the given system to an augmented matrix, and find all solutions by transforming to reduced echelon form and using back substitution.

Identify the row operation which transforms the matrix on the left to the matrix on the right.