For example:
If,, and are variables, and are scalars,
The following equation will be a linear combination:
Now put it in a linear algebraic environment. Our variable is now vector:, and the sum is the linear combination of variable scalar and vector to form a new vector:
A linear combination can only be added once or more. The linear combination of vector and scalar is generally written as:
What is span?
If the expansion of is expressed as:. For example, the following three vectors:
To prove this, we use a random vector.
One vector can be generated by simple mathematical linear combination with another vector, for example:
When a vector can be defined as a linear combination of other vectors, they are a group of linearly related vectors. When each vector in a set of vectors cannot be defined as a linear combination of other vectors, they are a set of linearly unrelated vectors.
In our example:
The simplest way to judge whether a set of vectors is a linear correlation set is to use determinant.
The linear combination of vector and scalar will extend the following important concepts: linear equations. We will only introduce two equations and two-variable equations in depth. In the wider course of linear algebra, you will learn more about the system of equations of n linear equations, where n can be any number.
Suppose there are two variables:
：
Now that we know how to multiply a vector by a scalar, let's calculate:
The above equation will lead to two independent equations:
The above equation is called binary quadratic equation. These equations can be solved by three theoretical methods:
Below, we will introduce these three methods in detail.
Graphic method:
Draw these two lines (linear diagram) and find the intersection. The intersection point is the solution because it is the only point on these two lines that exists at the same time. In other words, it is the only point that satisfies these two equations.
It can be clearly seen that:
Alternative method:
Separating a variable from one equation and then replacing it in the second equation will simplify the equation set to an equation with only one variable. In our example:
( 1)
(2)
From (1)
Replace b in equation (2)
And solve it.
Draw a conclusion after using simple algebraic knowledge, and substitute the above equation (1) or (2) to get.
Exclusion method:
In this method, we will eliminate one of the variables, that is, the absolute value of a scalar (coefficient), by multiplying the same number. Let's look at these two equations again:
If both sides of Equation 2 are multiplied by 5, the following equation will be obtained:
(Note: The scalar of the multiplication of two equations, that is, the absolute value of the coefficient, is 5. )
Now add them up. The following single equation with one variable is obtained:
Or:
Substitute into any equation to get.
The final answer:
.
The only solution is: