# Linear Combination

## Linear Combination

Sum of the products of the elements in a set V of mathematical objects by the elements in a set S of scalars.

### Example

If the variables x and y belong to a set of real numbers and a and b are integers, then the expression

$$z = ax + by$$

represents the real number z in the form of a linear combination of the integers a and b.

### Educational Note

The concept of a linear combination refers to two sets V and S of mathematical objects, which are vectors in a vector space and numbers or scalars in a numerical space, such as the set of real numbers. We define an external operation so that every element of V can be expressed as a sum of the products of a scalar of S and a vector of V. The number of terms in this product depends on the dimension of the chosen vector space. If the vector space is 2-dimensional, then each linear combination will include 2 terms; if the vector space is 3-dimensional, then each linear combination will include 3 terms; and so on.