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)
A is represented by B, A is irrelevant, A's rank is smaller than B
n+1 n-dimensional vectors are linearly related
Two equivalent linearly independent vector groups must contain the same number of vectors
The maximal irrelevant group of 3 vector groups contains the same number of vectors
4 row rank equals column rank equals rank
Elementary transformations do not change the rank
Rank is equal to the number of non-zero rows in the ladder matrix
The maximum independent group after elementary row transformation is equal to a non-zero column vector
Square matrix linear correlation--determinant=0
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
7 The rank of the augmented matrix = the rank of the coefficient matrix - the system of equations has a solution
Determination of solutions to linear equations
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