Linear algebra is a branch of algebra, which mainly deals with linear relations. Linear relationship means that the relationship between mathematical objects is expressed in linear form. For example, in analytic geometry, the equation of a straight line on the plane is a binary linear equation; The equation of spatial plane is a ternary linear equation, and the spatial straight line is regarded as the intersection of two planes, which is represented by an equation group composed of two ternary linear equations. A linear equation with n unknowns is called a linear equation. A function whose variable is once is called a linear function. Linear relation problem is called linear problem for short. The problem of solving linear equations is the simplest linear problem.
Linearity refers to the proportion and linear relationship between quantities, which can be mathematically understood as a function with a constant first derivative.
Nonlinear refers to the non-proportional nonlinear relationship, and the first derivative is not constant.
Determinant nonzero matrix reversible square matrix full rank vector group full rank (the number of vectors is equal to the dimension).
2. Determinants
2. 1 definition
The determinant det of a matrix is a scalar calculated based on the row and column data contained in the matrix. It is used to solve linear equations.
2.2 second-order determinant
Calculation method: diagonal rule
2.3 third-order determinant
Calculation method: diagonal rule
2.4 n-order determinant 2.4. 1 Calculate the reverse order number of permutation
2.4.2 calculating the determinant of order n
Simplified calculation summary
2.4.4 Three representations of determinant
2.5 the nature of determinant
The determinant of attribute 1 is equal to its transposed determinant.
Note: the rows and columns in the determinant have the same state, and the properties of the determinant are also true for the rows and columns.
Property 2 interchanges the two rows (columns) of the determinant, and the determinant changes sign.
Inference If two rows (columns) of a determinant are exactly the same, then the determinant is zero.
Property 3 All the elements in a row (column) of a determinant are multiplied by the same multiple k, which is equal to the determinant multiplied by the number k. 。
It is deduced that the common factor of all elements in a row (column) of a determinant can be mentioned outside the determinant symbol.
If the determinant of property 4 has two rows (columns), the determinant is zero.
Property 5 If the element in a column (row) of a determinant is the sum of two numbers, it is equal to the sum of the corresponding two determinants.
Property 6 Multiplies the elements in one column (row) of the determinant by the same multiple, and then adds them to the corresponding elements in another column (row), and the determinant remains unchanged.
2.6 Calculation method of determinant
1) usage definition
2) The value of determinant can be calculated by converting determinant into upper triangular determinant by using properties.
There are three conclusions in the theorem:
1) equations have solutions; (Existence of solution)
2) The solution is unique; (Uniqueness of Solution)
3) The solution can be given by Formula (2).
Theorem 4 If the coefficient determinant of a linear system of equations (1) is not equal to zero, then the linear system of equations must have a solution and the solution is unique.
Theorem 4' If a linear system of equations has no solution or two different solutions, its coefficient determinant must be zero.
Related theorems of homogeneous linear equations
Theorem 5 If the coefficient determinant d of homogeneous linear equations is not equal to 0, the homogeneous linear equations have only zero solutions and no non-zero solutions.
Theorem 5' If a homogeneous linear system of equations has a nonzero solution, its coefficient determinant must be zero.
1. Two Conditions for Solving Linear Equations by Cramer's Rule
1) The number of equations is equal to the number of unknowns;
2) The coefficient determinant is not equal to zero.
2. The significance of Kramer's law lies in establishing the relationship between the solution of linear equations, known coefficients and constant terms. Mainly suitable for theoretical derivation.
2.8 determinant is expanded by row (column)
The diagonal rule only applies to second-order and third-order determinants.
This section mainly considers how to use low-order determinant to represent high-order determinant.
3.[ number] matrix
3. Definition of1matrix
3. The difference between1.1matrix and determinant
3.2 Special Matrix
3.3 Matrix and Linear Transformation
3.4 Matrix operation 3.4. 1 matrix addition
Comparison between determinant and matrix addition;
Multiplication matrix
3.4.3 Matrix and Matrix Multiplication
Transposition of matrix
skew symmetric matrix
3.4.5 Determinant of Square Matrix
adjoint matrix
* * * Yoke matrix
3.5 invertible matrix (or nonsingular matrix)
3.6 Matrix Block Method
The block matrix is transposed not only in form, but also for each sub-block.
4. Elementary transformation of matrix and linear equations
4. Elementary transformation of1matrix
4.2 Equivalence relationship between matrices
4.3 the relationship between elementary transformation and matrix multiplication
4.4 Rank of Matrix
4.5 Multiple Solutions of Linear Equation
5. Linear correlation of vector groups
5. 1 vector group and its linear combination
5.2 Linear Correlation of Vector Groups
5.3 the level of vector group
Conclusion: The highest order non-zero subformula of a matrix is generally not unique, but the rank of the matrix is unique.
5.4 Structure of Solutions of Linear Equations
Question: What is the structure of the solution of linear equations?
A: The structure of solutions of linear equations is the relationship between solutions when there are infinite solutions of linear equations.
Remarks:
1) When a system of equations has a unique solution, there is no need to discuss the structure of the solution.
2) The following discussion assumes that the linear equations have solutions.
5.5 vector space 5.5. 1 closed concept
Definition: The so-called closure means that the result of the operation of any two elements in the set still belongs to the set.
5.5.2 The concept of vector space
Definition: Let V be a set of N-dimensional vectors, if
① the set v is not empty,
② The set V is closed for vector addition and multiplication,
Specifically, it is:
If a ∈ V and b ∈ V, then a+b ∈ v. (close to addition)
If a ∈ V, l ∈ R, then l a ∈ v. (closed for multiplier)
Then the set v is called a vector space.
5.5.3 The concept of subspace
Definition: If the nonempty set V 1 of vector space V is closed for the addition and multiplication operations defined in V, it is said that V 1 is a subspace of V. 。
5.5.4 Concept of Vector Space Basis
6. Similarity matrix and quadratic form
6. Inner product, length and orthogonality of1vector
6. 1. 1 inner product of vector
6. Length or norm of1.2 vector
Unit vector: a vector with the length of 1
6. Orthogonality of1.3 Vector
Vector orthogonality: vector inner product is 0.
6. 1.4 orthogonal matrix or orthogonal matrix
6. Properties of1.5 Orthogonal Matrix
6.2 eigenvalues and eigenvectors of square matrices
6.2. 1 positive definite matrix/semi-positive definite matrix
1) matrix is semi-positive definite if and only if each eigenvalue is greater than or equal to zero (>; =0)。
2) A matrix is positive if and only if each eigenvalue is greater than zero (>; 0)。
6.3 Similarity matrix
6.4 Diagonalization of Symmetric Matrix
6.5 Quadratic type and other standard types
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