What is the complex vector space and what is the relationship between complex numbers and vectors

Let f be a domain. The vector space on an f is a set v and two operations:

Vector addition: + v v v is denoted as v + w, v, w v

Scalar multiplication: · f v v is denoted as a v, a f, and v v

The following axioms are met ( a, b f and u, v, w v):

Vector additive associative property: u + v + w) = (u + v) +w;

Commutative law of vector addition: v + w = w + v;

The unit element of vector addition: v has a 0 called a zero vector in v, v v , v + 0 = v;

The inverse elements of vector addition: v v, w v such that v + w = 0;

Scalar multiplication is assigned to vector addition: a(v + w) = a v + a w;

Scalar multiplication is assigned to domain addition: (a + b)v = a v + b v;

Scalar multiplication is consistent with scalar domain multiplication: a(b v) = (ab)v;

Scalar multiplication has a unit element: 1 v = v, where 1 is the unit of multiplication of the domain f.

The elements of the complex vector space belong to the complex number field.

It's a plane, with only one more algebraic relation, i 2 = -1

Look at the textbook, there are detailed instructions.

The inflection scale is a representation of complex numbers, and it can only be a two-dimensional vector, that is, a plane vector. Complex numbers are limited to two-dimensional planes only. Complex numbers correspond one-to-one to vectors on complex planes starting from the origin.

1. Vector: In mathematics and physics, the quantity that has both magnitude and direction is called vector, also known as vector, which corresponds to quantity in mathematics and scalar quantity corresponding to its crack base height in physics;

2. Complex numbers: defined as pairs of binary ordered real numbers. The complex number field is an algebraic closure of the real front mask field, i.e., any complex coefficient polynomial always has roots in the complex number field.

Complex numbers were first introduced in the 16th century by the Italian Milanese scholar Cardan, and through the work of d'Alembert, Demoff, Euler, Gauss, and others, the concept gradually became accepted by mathematicians.

Vectors are a representation of complex numbers, and only.

Energy is a two-dimensional vector (plane vector). Vector also.

There are many other things that can be done, but only in the plural.

Constrained to a two-dimensional plane.

Strictly speaking, the origin is used in complex numbers and complex planes.

The vectors of the starting point correspond one-to-one.

Complex family vectors can be equal, but only if they are of the same magnitude and direction. In mathematics, a complex vector is a vector that consists of two real numbers, called the real part and the imaginary part, which represent the magnitude and direction of a complex number (real number) and an imaginary number (imaginary number). In the operation of complex vectors, arithmetic operations such as addition, subtraction, multiplication, and division of real and imaginary numbers can be used.

The condition for two complex vectors to be equal is that they are equal in both real and imaginary parts, i.e., they are of the same magnitude and direction. Therefore, two vectors of complex numbers can only be equal if they are equal in magnitude and direction.

The parallelism of complex vectors is an important concept, which means that two complex vectors are of the same direction but of different magnitude, or two beats of complex vectors with one complex vector of the same magnitude but different directions. If two complex vectors are in the same direction but of different sizes, they cannot be equivalent because they are of different sizes. If two vectors of complex numbers are of the same magnitude but in different directions, they are also not equivalent because they are in different directions.

1 Complex vectors can be equal.

2 In the complex trembling mu vector, in addition to the value of the element needs to be equal, the corresponding element position also needs to be equal.

That is, if vector a and vector b are equal, then their 1st element, 2nd element, 3rd element. The nth element needs to be equal.

3 Two complex vectors are considered equal when they are equal in both their values and positions.

The same conclusion applies to vectors of real numbers or any other number field.

It can be said that complex numbers and vectors have similar properties in some ways, while a complex number can be understood as a two-dimensional vector. But complex numbers and vectors are not exactly the same concept, so they cannot be considered two equal concepts.

A complex number is a number that is made up of real and imaginary parts and has specific rules of operation, such as addition, subtraction, multiplication, and division, etc. Vectors, on the other hand, are usually represented by an ordered array or coordinate point, and also have properties such as the length and direction of the vector.

When comparing complex numbers and vectors, it is necessary to pay attention to the differences in their operation rules and basic properties. Although a part of a complex number can be understood as a component of a vector, the real part and the imaginary part must be operated separately when adding and subtracting, while the vector operation is directly on the coordinates of the vector. Silver liquid pure.

Therefore, while complex numbers and vectors have quasi-sharp properties, they cannot be considered equivalent concepts.

1.The dimensionality of two complex vectors is the same, that is, the number of complex numbers contained is equal;

2.For each complex number, the real part of the hand cavity and the imaginary part are equal.

For example, if there are two complex vectors A and B, both of which have a dimension of 3, then the two vectors are equal to Bihler, and the following conditions need to be met:

A = A1 + B1i, A2 + B2i, A3 + B3I]B = C1 + D1I, C2 + D2I, C3 + D3I] where A1, B1, A2, B2, A3, B3, C1, D1, C2, D2, C3, D3 are real numbers, and I is an imaginary unit. Then if A and B are equal, the following conditions need to be met:

a1 = c1, b1 = d1

a2 = c2, b2 = d2

a3 = c3, b3 = d3

That is, for each complex number in each complex number vector, their real and imaginary parts are equal in a circle to be equal.

Complex vectors can not be changed bucket oak, etc., complex vectors correspond to real pairs, xy axis is the real number axis, but the complex vector x-axis is the real axis, y is the imaginary axis, the point on the vector is only the real part and the imaginary part of the pin complex number and not the complex number of the kernel itself, so the difference is very large.

No. It cannot be directly equal. It can only be said that the plane vector (1,1) can be represented by the complex number 1+i.

No comparison.

Because complex numbers are numbers of the form z=a+bi (a, b are real numbers) are called complex numbers, where a is called the real part, b is called the imaginary part, and i is called the imaginary unit. When the imaginary part of z is equal to zero, z is often called a real number; When the imaginary part of z is not equal to zero, and the real part is equal to zero, z is often called a pure imaginary number. A complex number field is an algebraic closure of a real number field, i.e., any complex coefficient polynomial always has roots in the complex number field.

Vectors (also known as Euclidean vectors, geometric vectors, vectors), refer to quantities with magnitude and direction. It can be visualized as a line segment with an arrow. The arrow points to:

represents the direction of the vector; Line Segment Length: Represents the size of the vector. The quantity corresponding to a vector is called a quantity (called a scalar in physics), and a quantity (or scalar) is only a magnitude and has no direction.

Related introductions. In the planar Cartesian coordinate system, two unit vectors i and j with the same direction as the x-axis and y-axis are taken as a set of bases. A is an arbitrary vector in the plane Cartesian coordinate system, and the coordinate origin O is the starting point and P is the end point as the vector A.

From the fundamental theorem of plane vectors, it can be seen that there is only one pair of real numbers (x,y) such that a=xi+yj, so the pair of real numbers (x,y) is called the coordinates of the vector a, and is denoted as a=(x,y). This is the coordinate representation of vector a. where (x,y) is the coordinates of the point p.

The vector a is called the position vector of the point p.

There are two types of modules in mathematics:

1. The modulus of complex numbers in mathematics. The value of the square root of the sum of the squares of the real and imaginary parts of a complex number is called the modulo of the complex number.

The algorithm for the two modules is as follows:

1. Let the complex number z=a+bi(a,b r).

then the modulo of the complex z |z|=√a^2+b^2

Its geometric meaning is the distance from a point (a,b) on a complex plane to the origin.

2. The function of the modulo operator " " is to find the remainder of the division of two numbers.

a%b, where both a and b are integers.

The calculation rule is that calculate a divided by b, and the resulting remainder is the result of modulo.

For example: 100% 17

So 100% 17 = 15

In this chapter, we will introduce some applications of complex numbers and vectors, especially in plane geometry. In addition, complex numbers will be used to solve the iterative problem of a class of functions, and the geometric meaning of complex numbers will be used to construct the interconnection between algebra and geometry, and the key point is how to select the appropriate coordinate system, and then establish the complex representation of geometric elements, so as to solve the plane geometry problem with the help of complex number operations.

1. Let the two points on the complex plane and the corresponding complex numbers be , then the distance between these two points is satisfied.

2. Let the complex numbers corresponding to the two points on the complex plane be , then the complex numbers corresponding to the fixed score point of the line segment can be expressed as.

3. Let the three points on the complex plane and the corresponding complex numbers be , and the sufficient and necessary conditions for these three points to be collinear are the existence of real numbers that are not all zero, so that the following two equations are true at the same time:

Fourth, let the four points that are not collinear and the corresponding complex numbers are , then , and the sufficient and necessary conditions for the four points of the contour are .

where is a non-zero real number.

5. Let the three points that are not collinear and the corresponding complex numbers be , respectively, then the area formula of is .