I. Determinants and matrices
The first chapter "determinant" and the second chapter "matrix" are the basic chapters in linear algebra, which need to be mastered skillfully.
The core content of determinant is to find determinant, including the calculation of concrete determinant and abstract determinant, in which there are two kinds of calculation of concrete determinant: low order and high order; The main method is to apply the properties of determinant and transform it into upper and lower triangular determinant through the row-column expansion theorem. For the evaluation of abstract determinant, the test site is not looking for determinant, but related properties. The matrix part is flexible, and the common knowledge points are operation rules, operation properties, determination and inversion of matrix invertibility, properties of matrix rank, properties of elementary matrix and so on.
Second, the vector and linear equations
Vector and linear equations are the core contents of the whole linear algebra. In contrast, determinant and matrix can be regarded as the basic chapters to discuss some problems of vector and linear equations; The contents of eigenvalues, eigenvectors and quadratic forms in the last two chapters are relatively independent, which can be regarded as the expansion of the core content.
Vector is closely related to the content of linear equations, and there are explicit or implicit associations between many knowledge points. The most effective way to review these two parts is to thoroughly straighten out the internal relations between many knowledge points, because doing so can first ensure real understanding and is also the premise of mastering and flexible application.
Solving linear equations can be regarded as the starting point and goal. Linear equation (general formula)
There are also two forms: (1) matrix form and (2) vector form.
1) Homogeneous linear equations are linearly related and uncorrelated.
Homogeneous linear equations can directly see that there must be a solution, because the equation must be established when all variables are zero; It is proved that a property of the vector part "zero vector can be expressed linearly by any vector".
Homogeneous linear equations must have solutions, which can be divided into two cases: ① there is zero solution; ② There is a nonzero solution. When the homogeneous linear equations have zero solutions, it means that the equations can only be established if the variables in the equations are all zero, while when the homogeneous linear equations have non-zero solutions, the above equations can only be established if the variables are not all zero. But the definition of judging whether the vector group is linearly related in the vector part is also based on this equation. So there is a connection between vectors and linear equations: whether the homogeneous linear equations have non-zero solutions and whether the column vectors of the coefficient matrix are linearly related. As you can imagine, the concept of linear correlation independence was put forward to better discuss the problems of linear equations.
2) The relationship between the solution of homogeneous linear equations and the largest independent group of rank sum.
It can also be considered that the introduction of rank is to better discuss linear correlation and linear independence. The definition of rank is "the number of vectors in the largest linearly independent group". Through the logical chain of "rank → linear correlation and irrelevance → determination of the solution of linear equations", it can be determined that when the column vectors are linearly correlated, the homogeneous linear equations have non-zero solutions, and the solution vectors of homogeneous linear equations can be expressed linearly by R linearly independent solution vectors (basic solution system).
3) Relationship between nonhomogeneous linear equations and linear representation
Whether a non-homogeneous linear equation group has a solution corresponds to whether a vector can be expressed linearly by a set of column vectors, and a set of numbers that make the equation hold is the solution of the non-homogeneous linear equation group.
Three. Eigenvalues and eigenvectors
Compared with the previous two chapters, this chapter is not the theoretical focus of linear algebra, but it is an examination focus. The reason is that a lot of contents in line generation need to solve related problems-determinant, matrix, linear equations, linear correlation, "pulling one hair and moving the whole body". The main points of this chapter are as follows:
1. The definition and calculation method of eigenvalues and eigenvectors is to remember a series of formulas and properties.
2. Similarity matrix and its properties, we need to distinguish the similarity, equivalence and contract of matrix:
3. The conditions for diagonalization of matrices also include two necessary and sufficient conditions and two sufficient conditions. The necessary and sufficient condition 1 is that the n-order matrix has n linearly independent eigenvalues; Necessary and sufficient condition 2 is that any r-multiple feature root corresponds to r linearly independent feature vectors.
4. Real symmetric matrix and its similar diagonalization. The real symmetric matrix of order n must be orthogonal to the diagonal matrix.
quadratic form
The content of this chapter is basically an extension of the fifth chapter "Eigenvalue and Eigenvector", because the core knowledge of the quadratic canonical form is that "the real symmetric matrix A has an orthogonal matrix C, so that A can be similarly diagonalized", and its process is the application of similar diagonalization in the previous chapter.
The main points of this chapter are as follows:
1. Quadratic form and its matrix representation.
2. Transform quadratic form into standard form by orthogonal transformation.
3. Judgment and proof of positive definite quadratic form.
Attached:
The first chapter determinant
1, the definition of determinant
2. The nature of determinant
3, the value of the special determinant
4. Determinant expansion theorem
5. Calculation of abstract determinant
Chapter II Matrix
1, definition of matrix and linear operation
Step 2 increase
3. Power of matrix
4. Move the item
5. The concept and properties of inverse matrix
6. Adjoint matrix
7, block matrix and its operation
8. Elementary transformation of matrix and elementary matrix
9. Equivalence of matrices
10, the rank of the matrix
Chapter III Carrier
The concept and operation of 1, vector
2. Linear combination and linear representation of vectors
3. Equivalent vector group
4. The linear correlation of vector groups has nothing to do with linearity.
5. Ranks of Maximal Linear Independent Groups and Vector Groups
6. Inner product and Schmidt orthogonalization
7.n-dimensional vector space (math 1)
Chapter IV Linear Equations
1, Cramer's rule for linear equations
2. Criteria for homogeneous linear equations with nonzero solutions
3. Existence criteria of solutions for nonhomogeneous linear equations.
4. Structure of solutions of linear equations
Chapter V Eigenvalues and Eigenvectors of Matrices
1, the concepts and properties of eigenvalues and eigenvectors of matrices
2. The concept and properties of similarity matrix.
3. Similar diagonalization of matrices
4. Eigenvalues, eigenvectors and similar diagonal matrices of real symmetric matrices.
Chapter VI Quadratic Form
1, quadratic form and its matrix representation
2. Contract transformation and contract matrix
3. Rank of quadratic form
4. Standard form and standard form of quadratic form
5. Inertia theorem
6. Transform quadratic form into standard form by orthogonal transformation and matching method.
7. Positive definite quadratic form and its judgment