Knowledge structure of advanced algebra

Knowledge structure of advanced algebra

One 、 Knowledge structure diagram of Higher Algebra

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Two 、 The content of knowledge structure of Higher Algebra

（ One ） linear algebra ：

Tools ： System of linear equations

1. determinant ：

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Multiply this determinant .

nature 3. If a row is the sum of two sets of numbers , Then this determinant is equal to the sum of two determinants , Except for this row, the two determinants are the same as the corresponding row of the original determinant .

nature 4. If two rows in a determinant are the same , So the determinant is zero .（ The two lines are the same, that is, the corresponding elements of the two lines are the same )

nature 5. If two rows in a determinant are proportional . So the determinant is zero .

nature 6. Add multiples of one line to another , The determinant remains unchanged .

nature 7. For the position of two rows in the newline column , Determinant inverse sign .

2. matrix ：

a. The rank of a matrix ： matrix A The number of non-zero rows in the matrix is called the rank of the matrix .

b. Matrix operation

Definition Homomorphic matrices ： It means that the number of rows corresponding to two matrices is equal 、 The corresponding matrix with the same number of columns ．　

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The equivalent transformation forms of matrix mainly include the following ：

1） Matrix i That's ok （ Column ） And j That's ok （ Column ） The position of is interchanged ;

2） With a non-zero constant k Multiply the number of matrix i That's ok （ Column ） Each yuan of ;

3） Put the second part of the matrix j That's ok （ Column ） All the yuan of k Times to the... Th i That's ok （ Column ） The corresponding yuan of .

3. System of linear equations

General linear equations . The general form of linear equations referred to here is

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a. Solution of linear equations

1) Elimination method

In elementary algebra , We have learned to solve simple bivariates by substituting into and subtracting from elimination 、 Ternary linear equations . actually , This method is more universal than using determinant to solve equations . But for those linear equations with high elements , The elimination method is quite tedious , Not easy to use .

2) Apply Clem's law

For the case that the number of unknowns is equal to the number of equations , We have

Theorem 1 If it contains

It's an equation

A system of elementary linear equations

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4. Vector correlation

a. The method of judging the linear correlation of vector groups

1) Linear correlation

2) The corresponding components of are proportional and linearly correlated

3) Vector groups containing zero vectors are linearly correlated

4) Vector groups are linearly correlated. At least one vector in this group can be linearly expressed by the rest of the vectors 5) Partial correlation is overall correlation

6) Let vector group be linearly expressed by vector group , If r>s, Then linear correlation ;

7)n+1 individual n Dimensional vectors must be linearly correlated ( The number is greater than the dimension )

8) The rank of the vector group is less than the number of vectors it contains, and the vector group is linearly correlated

9)n individual n The determinant formed by the vector of dimension =0 The vector group is linearly correlated

10) Each vector in the linear correlation vector group is also correlated after truncation

b. The method of judging the linear independence of vector group

1) Linearly independent

2) The corresponding component of is not proportional Linearly independent

3) The vector group is linearly independent. No vector in this group can be linearly expressed by other vectors

4) If the whole is irrelevant, the part is irrelevant

5) Each vector in the linear independent vector group is independent after lengthening

6) The rank of the vector group is equal to the number of vectors it contains The vector group is linearly independent

7)n individual n The determinant formed by the vector of dimension 0 The vector group is linearly independent

( Two ) Central topic ： Linear canonical

1. Quadratic type Linear flow pattern ：

Quadratic form and its matrix representation

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