A linear combination is a mathematical concept that is used in a variety of fields, including physics, engineering, and economics. It is a fundamental building block of linear algebra, and is combination of vectors. In this article, we will dive into the basics of linear combinations, including what they are, how they are used, and some examples of common applications.

What is a Linear Combination?

Think about the cartesian coordinates, you can represent any vector by giving different values to the X, Y and Z coordinates. Whenever you have different vectors you could also add them or subtract them to get a resulting vector out of them, you could also multiply them by numbers in order to give any of the vectors more 'weight' on the sum, the combination of multiplying the vectors by a certain value and adding them is called a linear combination. An easy way to think about it is just like a polynomial, where the constants are the vectors and the numbers that multiply them are the coordinates.

For example, if we have two vectors, A and B, and two scalars, x and y, the linear combination of A and B would be:

A + xB

This is known as a linear combination of A and B, with coefficients x and y.

How are Linear Combinations Used?

Linear combinations can be used whenever something can be represented by vectors, some of the applications are:

• Physics: You can model different forces acting on objects as linear combinations in order to make predictions over their movement.

• Programming: Sometimes linear combinations can be used to model graph problems or serve as a way to model a programming problem with polynomials, another popular example of how linear combinations and polynomials are related to programming is the question of whether P=NP or not.

• Engineering: Just like in physics, it´s possible to make several different simulations involving linear algebra, such as the drag or lift of an airplane based on flow turbulence or the movement of an Elevator with people on it.

• Economics: It´s possilbe to transform a weighted average into a linear combination of vectors and with that make calculations of supply and demand or compare different countries GDP's and things of that nature.

Conclusion

In conclusion, linear combinations are a fundamental concept in mathematics that are used in a variety of fields, including physics, programming, engineering, and economics. They are used to represent forces, velocities, and electric fields. Understanding the basics of linear combinations is essential for anyone working in these fields and will help you to better understand the underlying concepts.