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System of linear equations
This is a mind map about systems of linear equations, which are algebraic equations in one or more variables where each variable has an exponent of 1 (i.e. they do not have squares, cubes, etc.).
Edited at 2024-03-02 17:35:49- bacteria
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System of linear equations
- bacteria
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
- Plant asexual reproduction
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
- Reproductive development of animals
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
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- Outline
Advanced Algebra 1
System of linear equations
definition
1Elementary Transformation of Equations
elimination method
2 ordered array (vector)
3-7 Vector calculation
3 vectors are equal
4 vector addition
5 zero vector
6 subtraction
7 numbers multiplication
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
theorem
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.
elimination method
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
linear correlation
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
rank
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