Solving systems of linear equations
This week, you will learn how to solve a system of linear equations using the elimination method and the row echelon form. You will also learn about an important property of a matrix: the rank. The concept of the rank of a matrix is useful in computer vision for compressing images.
Learning Objectives
- Solve a system of linear equations using the elimination method.
- Use a matrix to represent a system of linear equations and solve it using matrix row reduction.
- Solve a system of linear equations by calculating the matrix in the row echelon form.
- Calculate the rank of a system of linear equations and use the rank to determine the number of solutions of the system.
Solving systems of linear equations: Elimination
Solving non-singular systems of linear equations
If two equations are true, then so are the calculations
a = 3, b = 8
Correct! The system of equations has a unique solution at point (3, 8), meaning that a = 3, b = 8. You can verify these values by replacing them in the equations.
Solving singular system of linear equations
The system above has 1 degree of freedom because the variable "a" can be any number and "b" is a constant minus "a"
Since it has 1 variable that can be any number it has 1 degree of freedom
Infinitely many solutions
Equation 2 is (2 ⋅ Equation 1), meaning there are infinitely many solutions.
Solving a system of equations with more variables
Matrix row-reduction
In the intermediate system (aka row echelon form), we get b = 2 by using the elimination method and solving for b
The reduced row echelon form is the solved system represented as a matrix
We get the second equation as 0a + 0b = 0 since we can't solve the system
3 rules for row echelon form:
- Diagonal numbers can be 0 or 1
- The lower part of the diagonals is always zero
- The upper part of the diagonals can be any number or 0 based on:
- If the diagonal number is 1, then it can be any number
- If the diagonal number is 0, then it’s 0
Row operations that preserve singularity
Rules used in linear equations can also be applied to matrices
Following manipulations preserve the singularity and non-singularity of the matrices
All the information here is based on the Linear Algebra for Machine Learning and Data Science | Coursera from DeepLearning.AI
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