1Elementary Transformation of Equations

2 ordered array (vector)

3-7 Vector calculation

8n-dimensional vector space

n-dimensional vector space

9 Linear combination, linear expression (single vector and vector group)

10 Linear combinations (between vector groups)

11Linear correlation

12 Linearly independent

13 extremely irrelevant group

14 ranks are defined using maximal independent groups

linear correlation

15 Row Rank and Column Rank

16 sub-forms, main sub-forms, sequential main sub-forms

17 rank is defined by the highest order non-zero subformula

rank

1. If the number of equations in a system of homogeneous equations is less than the number of unknowns, there must be a non-zero solution.

2 More is expressed by less, and more is related (more refers to the group with more vectors)

The maximal irrelevant group of 3 vector groups contains the same number of vectors

linear correlation

4 row rank equals column rank equals rank

5 Homogeneous solution condition only has 0 - coefficient matrix determinant = 0; non-zero solution condition - coefficient matrix determinant ≠0

6 Non-homogeneous equations have non-zero solutions - coefficient matrix determinant = 0

rank

7 The rank of the augmented matrix = the rank of the coefficient matrix - the system of equations has a solution

8 The number of basic solution systems of a homogeneous equation system is the unknown minus the rank (the number of free unknowns)

9. The solution of a system of non-homogeneous equations is a special solution + derived group basic solution system

Structure of solutions to linear equations