Systems of Linear Equations

This chapter systematically studies the basic concepts of systems of linear equations, structure of solutions, Cramer’s rule, fundamental solution sets, general solutions, and elementary row transformations.

Basic Concepts of Systems of Linear Equations

• General form: $A\vec{x} = \vec{b}$
• Homogeneous system: $A\vec{x} = 0$
• Nonhomogeneous system: $A\vec{x} = \vec{b}$

Structure of Solutions

• Homogeneous: zero solution, nonzero solution, fundamental solution set, general solution
• Nonhomogeneous: particular solution + general solution of the homogeneous system

Cramer’s Rule

• Unique solution if $|A| \neq 0$, $x_i = \frac{|A_i|}{|A|}$

Elementary Row Transformations and Gaussian Elimination

• Elementary row transformations: swap, add multiple, scalar multiplication
• Gaussian elimination method for solving systems

Fundamental Solution Set and General Solution

• Fundamental solution set: maximal linearly independent set of solutions to the homogeneous system
• General solution: linear combination of the fundamental solution set

Exercises

1. Determine the structure of solutions for the system $\begin{cases} x + y = 2 \\ 2x + 2y = 4 \end{cases}$.
2. Use Cramer’s rule to solve $\begin{cases} x + 2y = 5 \\ 3x - y = 4 \end{cases}$.
3. Write the fundamental solution set for $\begin{cases} x + y + z = 0 \\ 2x + 2y + 2z = 0 \end{cases}$.
4. Use Gaussian elimination to solve $\begin{cases} x + y + z = 6 \\ 2x + 3y + z = 13 \\ x - y + 2z = 4 \end{cases}$.
5. Determine whether $\begin{cases} x + y = 1 \\ x + y = 2 \end{cases}$ has a solution.