Lecture 12: Maximal Linearly Independent Sets and Rank

12.1 Maximally Linear Independent Groups

12.1.1 Basic Concepts of Maximally Linear Independent Groups

#####Definition: Maximallylinearindependentgroup

description: $$\text{ In the set of vectors }a_1,a_2,\cdots,a_s\text{ If there are subgroups, }a_{i_1},a_{i_2},\cdots,a_{i_r}\text{ Satisfaction }:$$

$$1.a_{i_1},a_{i_2},\cdots,a_{i_r}\text{ linearly independent }$$

$$2. \text{ Any vector in a set of vectors } a_i \text{ is the maximal linearly independent set of the original vector set }$$

is called $$a_{i_1},a_{i_2},\cdots,a_{i_r}\text{ is a maximal linearly independent set of the protovector group }$$

Explanation

• Concept:
• Maximally linearly independent sets of vector sets are generallynot unique;
• A set of vectors composed of only one zero vector does not have a maximally linear-irrelevant group; the maximally linearly independent group of a linearly independent group is the set itself;
• $a_{i_1},a_{i_2},\cdots,a_{i_r}$
• From the original vector set;
• Select some groups from the vectors: $i_r$ come out;
• Note:
• Maximally linearly independent sets are not unique;
• Zero vectors are not studied;

12.1.2 Equivalent Vector Sets

#####Definition: Equivalencevectorset

description: In $\text{ Let there be two sets of vectors: (I) }\alpha_1,\alpha_2,\cdots,\alpha_s,(II)\beta_1,\beta_2,\cdots,\beta_i.\text{ If every vector in (I). }\alpha_i(i=1,2,\cdots,s)\text{ All can be passed through }$ (II), vectors are represented linearly, so the set of vectors (1) can be represented linearly by the set of vectors (II); If sets (I) and (II) can be represented linearly with each other, then the set of vectors (I) and (I) are called equivalent sets of vectors, denoted as (I) similar to (II).

Explanation

• Concept:
• Representsmultiple vectors can be represented by multiple vectors;
• Equivalence vector sets are not maximally linearly independent groups; there may be "lying flat" sets of vectors;
• So it's not "equal," but rather calledequivalent;
• Purpose:
• Essentially, it means:The space formed by the set of equivalent vectors is consistent;
• Explanation:
• A and B are equivalent <-> A can be represented by B + B can be represented by <-> r(A)=r(B)=r(A|B)

Supplement: Conditions for fulfillment

• $r(\alpha_{1}\alpha_2,\cdots,\alpha_s)=r(\beta_1,\beta_2,\cdots,\beta_i).$
• Proof:
• Not goodthere is/are:$r(\alpha_{1}\alpha_2,\cdots,\alpha_s,\beta_1,\beta_2,\cdots,\beta_i)=r(\beta_1,\beta_2,\cdots,\beta_i)=r(\alpha_{1}\alpha_2,\cdots,\alpha_s)$
• then it means: $\alpha_{1}\alpha_2,\cdots,\alpha_s$ can be represented by $\beta_1,\beta_2,\cdots,\beta_i$;
• Also means: $\beta_1,\beta_2,\cdots,\beta_i$ can also be represented by $\alpha_{1}\alpha_2,\cdots,\alpha_s$;
• Sufficient conditions for a set of equivalence vectors <- Equivalence of three ranks:
• $$r(\alpha_{1}\alpha_2,\cdots,\alpha_s,\beta_1,\beta_2,\cdots,\beta_i)=r(\beta_1,\beta_2,\cdots,\beta_i)=r(\alpha_{1}\alpha_2,\cdots,\alpha_s)$$
• Meaning: Transform it intothe problem of whether their ranks are the same;

Note: Candidates should pay attention to the differences between equivalence matrices and equivalence vector sets;

• Matrix equivalence must be isomorphic: A,B Equivalence -> r(A)=r(B).
• Example: $(\begin{matrix}1\\0\end{matrix})\text{ and }(\begin{matrix}0\\1\end{matrix})$

-Satisfactorycondition for an equivalence set of vectors: $r(A|B)=r(B)=r(A)$

12.2 Rank of a set of vectors

12.2.1 Basic Concepts of Rank in sets of vectors

#####Definition: Therankofthevectorset

description: $$r(\alpha_{1},\alpha_{2},\cdots,\alpha_{s})=r$$

Equivalence vector sets have equal ranks; the opposite may not hold

Explanation

• The dimensionality of the space for a maximally independent group;

12.2.2 Important Theorems and Formulas Related to Rank

#####Theorem: Threeranksareequal

Description: r(A) (matrix rank) = row rank of A (rank of A's row vector set) = column rank of A (rank of A's column vector set)

Explanation

• The ranks of sets of vectors viewed vertically and horizontally are the same;

#####Theorem: AArrivedBofElementary line transformation

description: if elementary row transformation is performed from A to B, then:

1. The row vector sets of A and B are equivalence vector sets;

2. Any corresponding subcolumn vector sets of A and B have the same linear dependence;

Explanation

• 1. The row vector sets of A and B are equivalent vector sets <- $r(A|B)=r(B)=r(A)$
• 2. Any corresponding subcolumn vector sets of A and B have the same linear dependence;
• Not only are the properties identical, but the final solution is also the same;

#####Theorem: Alinearrepresentationofasetofvectors

Description: if the set of vectors A can be represented by B, then $r(A)<=r(B)$

12.2.3 Problem Solving: Find a maximal linear independent set

Method: Steps to find a maximal linear independent set

• 1. Perform an elementaryrowtransformation on the matrix formed by column vectors, convert it into a row ladder matrix, and determine the r(A);
• 2. Find a submatrix of rank r(A) by column, which is a maximal linearly independent group;

12.2.4 Conclusion

Conclusion One: r(AB)≤min{r(A),r(B)

Conclusion Two: r(A+B)≤r([A, B]≤r(A)+r(B)

Conclusion 3: Let A be a n (n≥2)-order matrix, $A^*$ be the adjoint matrix of $A$, then:

• $$r\left(A^{*}\right)=\begin{cases}n,&r\left(A\right)=n,\\1,&r\left(A\right)=n-1,\\0,&r\left(A\right)<n-1.\end{cases}$$