The term linear means that something shows a constant evolution in a certain direction. This, in the field of physics and mathematics.
That is, linear means that there is a variation that will always be of the same magnitude, and in the same sense.
For example, imagine that the heating time in a microwave oven can be increased from 10 seconds to 10 seconds before pressing the start button. This means that the period for heating the food is adjusted linearly.
We must remember that linear equations are those equations of the first degree. That is, those where the variable is raised to the power one. Its general form, when they have two unknowns, is the following:
y = mx + b
In the example above, y is the dependent variable, x is the independent variable, and the coefficients are a and b.
This type of equations can be represented by a line, where m is its slope. In the same way, we can notice that x is the variable that goes on the horizontal axis, while y goes on the vertical axis and b is the point where the line intersects the vertical axis. We can see the example in the image below:
Another of the simplest forms of a first degree equation is when it has a single variable, so it can be expressed as:
c = ax + b
In the equation above, x is the unknown, which is multiplied by the coefficient (a), while b and c are constants.
The linear function is one where two conditions are met:
- Additive Property: If I have f (x) and f (y), then f (x) + f (y) = f (x + y).
- Homogeneous property: It is true that Af (x) = f (Ax). This, being A a natural number.
If these two properties are fulfilled, it is called the principle of superposition.
It should be noted that these principles are not always fulfilled in an equation of the first degree, only when the coefficient b is zero.
Linear algebra is the branch of mathematics that is dedicated to the study of elements such as matrices, vectors, vector spaces, and systems of linear equations.
Linear algebra is one of the most complex areas of algebra and is usually the field of study and application mainly of engineering and computer science.