Linear Systems of Equations

Linear Systems

• Focus is on systems of linear equations

Introduction to Linear Equations

• A system of linear equations involves finding common solutions.
• A linear equation system is a set of equations in the form:
  • $a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n = b_1$
  • $a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n = b_2$
  • $a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n = b_m$
  • $x_1, x_2,..., x_n$ represent the unknowns
  • $a_{ij}$ are coefficients
  • $b_i$ are constants.
• A solution is a set of values for $x_1, x_2,..., x_n$ that satisfy all equations in the system.
• A compatible system has at least one solution
• An incompatible system has no solutions

Matrix Representation

• Systems can be written in matrix form: $Ax = b$
  • $A$ is the coefficient matrix.
  • $x$ is the vector of unknowns.
  • $b$ is the vector of constants.
• For the system:
  • $2x_1 + 3x_2 = 5$
  • $x_1 - x_2 = 1$
• The matrix form is:
  • $\begin{bmatrix} 2 & 3 \ 1 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 5 \ 1 \end{bmatrix}$

Elementary Row Operations

• Operations on matrix rows include:
  • Swapping two rows: $L_i \leftrightarrow L_j$
  • Multiplying a row by a non-zero scalar: $L_i \leftarrow \lambda L_i$, where $\lambda \neq 0$
  • Adding a scalar multiple of one row to another: $L_i \leftarrow L_i + \lambda L_j$
• These operations simplify solving linear systems by transforming them into equivalent, easier-to-solve systems.

Solving Linear Systems Using Gauss Method

• The Gauss method systematically solves linear equations.
• Apply elementary row operations on the augmented matrix $[A|b]$ to transform it into a reduced echelon form.
• Requirements for a matrix to be in reduced row echelon form:
  • All zero rows are at the bottom.
  • The leading entry (pivot) in each non-zero row is 1.
  • Each pivot is to the right of the pivot in the row above it.
  • All entries above and below a pivot are zero.
• Example of Gauss method:
  • Solve:
    • $x_1 + x_2 + x_3 = 3$
    • $2x_1 + 3x_2 + x_3 = 5$
    • $3x_1 + x_2 - 2x_3 = -2$
  • Augmented matrix:
    • $\begin{bmatrix} 1 & 1 & 1 & 3 \ 2 & 3 & 1 & 5 \ 3 & 1 & -2 & -2 \end{bmatrix}$
  • Applying Gauss method leads to:
    • $\begin{bmatrix} 1 & 0 & 0 & \frac{2}{7} \ 0 & 1 & 0 & \frac{6}{7} \ 0 & 0 & 1 & \frac{13}{7} \end{bmatrix}$
  • Solution: $x_1 = \frac{2}{7}, x_2 = \frac{6}{7}, x_3 = \frac{13}{7}$

Existence and Uniqueness of Solutions

• A linear system can have one, infinite, or no solutions.
• The number of solutions depends on the rank of the coefficient matrix $A$ and the augmented matrix $[A|b]$.
• Rouché-Fontené Theorem states a linear system $Ax = b$ is compatible if and only if $rank(A) = rank([A|b])$.
  • If $rank(A) = rank([A|b]) = n$ ($n$ = number of unknowns), the system has a unique solution.
  • If $rank(A) = rank([A|b]) < n$, the system has infinitely many solutions.
  • If $rank(A) < rank([A|b])$, the system has no solution.

Homogeneous Systems

• A linear system is homogeneous if all constants $b_i$ are zero
• Represented as $Ax = 0$.
• Homogeneous systems are always compatible, with the trivial solution $x = 0$.
  • If $rank(A) = n$, the trivial solution is the only solution.
  • If $rank(A) < n$, the system has infinitely many solutions.

Exercises

• Exercise examples provided:
  • Solving systems using the Gauss method
  • Determining system compatibility
  • Finding the number of solutions based on a parameter.

Course Conclusion

• Covered the basics of linear equation systems.
• Matrix representation
• Gauss method for solving
• Discussion on the existence and uniqueness of solutions.