What is a System of Linear Equations - Linear Algebra

What is a System of Linear Equations – Linear Algebra

I know that you’re probably interested in machine learning, but did you know that linear algebra is a must to understand many algorithms? To deeply understand and tune them you need to be familiar with what’s under the hood.

In this series of articles on Linear Algebra, I will go through the main concepts so that you understand the fundamentals. To understand these articles you need to be familiar with algebra.

What is a Linear Equation?

A linear equation is simply an equation of the following form:

3x_{1}+5x_{2}+...+2x_{n}=b          [1]

I chose the coefficients randomly, so they could have been any real or complex numbers.

The following equation is not linear because of the square root:

3x_{1}+5x_{2}=2 \sqrt x_{3}          [2]

One of the simplest linear equation that you may recall from high school is:

mx+b=y          [3]

A linear equation in this context is similar, but can take more than 2 xs.

A System of Linear Equations

Now imagine two linear equations using the same variables. For example:

3x_{1}+5x_{2}=4          [4]

x_{1}-2x_{2}=2          [5]

What is a System of Linear Equations – Linear Algebra - A system of linear equations

Here x1 was treated as the independent variable, and x2 as the dependent variable. Here’s the Python code to create this chart:

Multiple linear equations who share the same variables is called a System of linear equations.

When having multiple linear equations it’s possible to find some values of xs that make all the equations true. For example, in the example above a solution is [ 0.90909091, 1.27272727]:

In this case there is a unique solution, but a System of linear equations can also have no solution, or an infinity of solutions. The set of all solutions is called the Solution Set.

Formally, we call a system with no solution incompatible, and a system with at least one solution compatible.

Matrix Notation

Writing equations like [4] and [5] becomes quickly tedious for bigger systems of linear equations, and so using the matrix notation is beneficial. We can resume the system in [4] and [5] like this:

\begin{bmatrix}3 & 5 & 4\\1 & -2 & 2\end{bmatrix}          [6]

Columns 1 and 2 are the coefficients on the left side of the equations [4] and [5], and the last column is the coefficients of the right side.

We can also look at the size of a matrix, which in this case is a 2 by 3 matrix. The first number is always the number of rows, and the second one is the number of columns.

Conclusion

Here’s what you should remember:

  • Multiple linear equations sharing the same variables are called a System of linear equations.
  • A system of linear equations can have no solution, one solution, of an infinity of solutions.
  • The matrix notation is beneficial for larger system of linear equations and also to perform calculations that we will see in future articles.

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