Understand various matrix operations, matrix decompositions, factorization and related operations
System of Linear Equations
A system of linear equations consists of multiple linear equations involving the same set of variables. Solving such systems helps determine the values of unknowns that satisfy all equations simultaneously.
Consider a system of \( m \) equations with \( n \) unknowns, represented in matrix form as:
\( A\vec{x} = \vec{b} \)
where:
- \( A \) is an \( m \times n \) coefficient matrix
- \( \vec{x} \) is an \( n \times 1 \) vector of unknowns
- \( \vec{b} \) is an \( m \times 1 \) constant vector
Example:
Suppose we have the following system:
\[
\begin{aligned}
x + 2y &= 5 \\
3x + 4y &= 11
\end{aligned}
\]
It can be written in matrix form as:
\[
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\begin{bmatrix}
x \\
y
\end{bmatrix}
=
\begin{bmatrix}
5 \\
11
\end{bmatrix}
\]
Solution Methods:
There are several techniques to solve such systems:
- Gaussian Elimination: Reduces the system to row echelon form and then back-substitutes to find the solution.
- Matrix Inversion: If \( A \) is square and invertible, the solution is \( \vec{x} = A^{-1}\vec{b} \).
- LU Decomposition: Factors \( A \) into \( LU \) and solves via forward and backward substitution.
Interpreting Solutions:
- Unique Solution: The system has one solution (e.g., lines intersect at a single point).
- Infinite Solutions: The equations are dependent (e.g., same line).
- No Solution: The system is inconsistent (e.g., parallel lines).
Systems of linear equations are fundamental in engineering, physics, and computer science for modeling real-world problems.