What Is a Vector Pack?

In mathematics, vectors (also called Euclidean vectors, geometric vectors, vectors) refer to quantities with magnitude and direction. It can be visualized as a line segment with arrows. Pointed by the arrow: represents the direction of the vector; line length: represents the size of the vector. The quantity corresponding to a vector is called quantity (called scalar in physics), and quantity (or scalar) has only size and no direction.

Vectors were originally applied to physics. a lot of

The value of the determinant is a number representing the [element] size of the space in which the vector is located.

For example, in a plane rectangular coordinate system, the entire plane can be composed of a square whose length and width are both 1. The size of this square is 1. This square is the [element] in the plane rectangular coordinate system, and its size is 1.

All points in the plane coordinate system can be used

These two vectors are used to describe, these two vectors are also called the "scale" of the plane rectangular coordinate space.

Determinant of these two vectors

Then, the size of the plane rectangular coordinate system cell is a square with a size of [element] 1.

For another example, we stretch the plane rectangular coordinate system and use the following two vectors to describe

Then, the [element] of this new coordinate system (2-dimensional space) is a rectangular block of size 2.

For another example, we deform the plane rectangular coordinate system and use the following two vectors to describe

Then, the [element] of this new coordinate system (2-dimensional space) is a parallelogram block of size 2.

From the above three examples, it can be seen that in a two-dimensional space, the value of the determinant composed of two two-dimensional vectors is equivalent to the size of the area of a parallelogram composed of two vectors. That is, in a two-dimensional space, the value of the determinant composed of two two-dimensional vectors is equivalent to the [cross product] of two two-dimensional vectors.

Further, look at 3D space.

For example, in a space rectangular coordinate system, this space can be composed of a cube whose length, width, and height are both 1. The size of this cube is 1. This cube is the [Element] in the space rectangular coordinate system (3D space), and its size is 1.

Then it can be seen that in a three-dimensional space, the value of the determinant composed of three three-dimensional vectors is equivalent to the [mixed product] of three three-dimensional vectors.

This expands to n-dimensional space. In the n-dimensional space, the value of the determinant composed of n n-dimensional vectors represents the [element] size of the n-dimensional space where the n-dimensional vector is located. At the same time, these n n-dimensional vectors are also called [scale] of n-dimensional space.

Assume

Vector definition

Given a field F, the vector space on F is an F-module.

Vector isomorphism

Given two vector spaces V and V 'on the field F, if there is a bijective : V V', and

. Then V and V 'are isomorphic.

Vector mapping

For two vector spaces V and W in the same F field, set a linear transformation or "linear mapping" from V to W. These mappings from V to W have in common that they maintain the sum and scalar quotient. This set contains all linear mappings from V to W, described by L (V, W), and is also a vector space in the F field. When V and W are determined, the linear mapping can be expressed by a matrix. Isomorphism is a one-to-one linear mapping. If there is isomorphism between V and W, we call these two spaces isomorphism. A vector space in the F-field plus a linear mapping can form a category, the Abel category.

Vector extension

Studying vector space generally involves some additional structure. The additional structure is as follows:

A real or complex vector space plus the concept of length. The norm is called the normed vector space.

The concept of a real or complex vector space plus length and angle is called the inner product space.

A vector space plus topological conformance operations (addition and scalar multiplication are continuous mappings) is called a topological vector space.

A vector space plus a bilinear operator (defined as vector multiplication) is a domain algebra.

Vector subspace and basis

A non-empty sub-set W of a vector space V is hermetic in addition and scalar multiplication. It is called a linear subspace of V. Given a vector set B, the smallest subspace containing it is called its expansion, and it is written as span (B). Given a vector set B, if its expansion is the vector space V, then B is called the generator set of V. The largest linear independent subset of a vector space V is called the basis of this space. If V = 0, the only basis is the empty set. For a non-zero vector space V, the basis is the generator set with the smallest V. If a vector space V has a generator set with a limited number of elements, then V is said to be a finite-dimensional space. All bases of a vector space have the same cardinality, which is called the dimension of the space. For example, a real vector space:

in,

The dimension is n. Each vector in space has a unique way to express it as a linear combination of the elements in the basis. By arranging the elements in the base, the vector can be represented in a coordinate system.