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System of Linear Equations | Linear Algebra
Mathematics for Data Analysis and Modeling
course content

Course Content

Mathematics for Data Analysis and Modeling

Mathematics for Data Analysis and Modeling

1. Basic Mathematical Concepts and Definitions
2. Linear Algebra
3. Mathematical Analysis

System of Linear Equations

A system of linear equations (SLE) is a set of equations where each equation is a linear combination of variables. The goal is to find a solution that satisfies all the equations simultaneously.

Example

Let's look at the example of a system of linear equations:

We have 3 unknown variables x, y and z and have 3 equations that include all of these variables.

Solving the system

How can we solve the system? Firstly, let's rewrite it in a matrix form:

Expressing the system of linear equations in matrix form provides us with a straightforward approach for solving the system utilizing the inverse matrix:

To use this approach we have to be sure that matrix A can be inversed:
The square matrix A can be inversed if and only if its determinant is nonzero.

Example 1

We have found the solution using the inverted matrix.

Example 2

The code above produces an error - the matrix is singular (has zero determinant) so we can't solve the system of equations.
The explanation for this is quite simple: the matrix rows are linearly dependent (the third row is the sum of the first two). As a result, the third equation doesn't provide any additional information, and we are left with a system of 3 variables but only 2 unique equations. As a result such a system either have no solutions or there are lots of solutions.

Everything was clear?

Section 2. Chapter 7
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