Linear combinations of independent variables

Concepts and Definitions

A linear combination is an expression built from a set of terms (variables or vectors) by multiplying each term by a constant and adding the results.
Vectors are said to be independent if there is no scalar combination of the vectors that will lead to the zero vector, unless all scalars are zero.
If vectors are not independent, they are dependent, and some vector in the set can be expressed as a linear combination of the others.
A vector space is the set of all possible linear combinations of a set of vectors.
A basis set is a set of vectors within a vector space that are linearly independent and span the space.

The linear combination of two independent variables is also an independent variable which can be expressed as a function of the original variables.
The Span of a set of vectors is the set of all possible linear combinations of the vectors.
Any set of p vectors spans a subspace of dimension at most p.
Whether a set of vectors spans a particular space is not just dependent on the number of vectors but also on their relative position.
If a set of vectors are linearly independent, there is no way to express any of the vectors as a linear combination of the others. This is equivalent to saying that the only solution to the equation a1v1 + a2v2 +…+ anvn = 0 is a1 = a2 =…= an = 0.

To find a linear combination of vectors, multiply each vector by a scalar (any real number) and add the results.
To check if a vector is a linear combination of other vectors, set up and solve a system of linear equations.
To check if vectors are linearly independent, set up and solve a linear system of equations and verify that the only solution is the trivial solution.
To find the span of a set of vectors, look for all the linear combinations of the vectors.
To determine if a set of vectors spans a space, set up and solve a system of linear equations for each vector in the given space.

Linear combinations play a crucial role in solving systems of linear equations and linear differential equations.
They are commonly used in mathematical modelling, linear programming, and vector calculus.
For instance, in physics the state of a quantum system can be represented as a linear combination of basis states. The same principle applies in the field of computer science in quantum computing.
In statistics, regression techniques often employ linear combinations to predict a dependent variable based on several independent variables.
In image processing, a digital image can often be viewed as a linear combination of base images, also called image bases.