**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Numerical Computation and Linear Algebra **Topic:** Solving linear systems of equations using matrix methods. ### 1. Introduction Linear systems of equations are fundamental in various fields, such as engineering, physics, economics, and computer science. MATLAB provides an efficient and effective way to solve these systems using matrix methods. In this topic, we will explore the different techniques and functions available in MATLAB to solve linear systems of equations. ### 2. Representing Linear Systems as Matrix Equations A linear system of equations can be represented as a matrix equation of the form: `Ax = b` where `A` is the coefficient matrix, `x` is the vector of unknowns, and `b` is the constant vector. For example, consider the following system of linear equations: `2x + 3y = 7` `x - 2y = -3` This system can be represented as the matrix equation: ```matlab A = [2 3; 1 -2]; b = [7; -3]; ``` ### 3. Solving Linear Systems using `mldivide()` (Backslash Operator) MATLAB provides the `mldivide()` function, also known as the backslash operator (`\`), to solve linear systems of equations. The syntax is: `x = A \ b` Using the example above: ```matlab x = A \ b ``` This will output the solution `x`. ### 4. Solving Linear Systems using `inv()` (Inverse Matrix) Another way to solve linear systems is by using the inverse matrix method: `x = inv(A) * b` However, this method is less efficient and accurate than the `mldivide()` method. ### 5. Solving Linear Systems using `linsolve()` (Linear System Solver) The `linsolve()` function is a more robust and efficient solver that can handle complex systems: ```matlab [x, r] = linsolve(A, b) ``` This function returns the solution `x` and the residual `r`. ### 6. Handling Linearly Dependent Systems When the coefficient matrix `A` is singular or has linearly dependent rows/columns, the system is ill-conditioned. In such cases, MATLAB will display a warning or error message. To handle such systems, use the `rank()` function to calculate the rank of the matrix: ```matlab rank(A) ``` If the rank is less than the number of rows/columns, the system is linearly dependent. ### 7. Handling Overdetermined Systems When the system has more equations than unknowns, it is overdetermined. In such cases, MATLAB will find the least-squares solution using the `mldivide()` function. ### 8. Key Concepts and Takeaways * Representing linear systems as matrix equations. * Solving linear systems using `mldivide()`, `inv()`, and `linsolve()`. * Handling linearly dependent and overdetermined systems. * Using `rank()` to determine the rank of a matrix. ### Example Code ```matlab % Define the coefficient matrix A and constant vector b A = [2 3; 1 -2]; b = [7; -3]; % Solve the linear system using mldivide() x = A \ b % Define an overdetermined system A_over = [1 2; 3 4; 5 6]; b_over = [7; 8; 9]; % Solve the overdetermined system using mldivide() x_over = A_over \ b_over % Check the rank of the matrix rank(A) ``` ### Additional Resources * [MATLAB Documentation: Solving Linear Systems](https://www.mathworks.com/help/matlab/math/solving-linear-systems.html) * [Linear Algebra for Everyone](https://www.khanacademy.org/math/linear-algebra) **Leave your comments, questions, or ask for help below this post.** Your feedback is invaluable in improving the quality of this course. **What's next?** In the next topic, we will explore **Eigenvalues, Eigenvectors, and Singular Value Decomposition (SVD)**.