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rank of vector group

The rank of a vector group is the number of vectors in the maximum linearly independent group in the vector group, that is, the number of the largest independent vectors in the vector group.
For example, for the set of vectors A = ((1, 2, 3), (4, 5, 6), (7, 8, 9)), its rank is 3 because it contains three linearly independent vectors.
For the vector group B = ((1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12)), its rank is also 3, because any three of them The vectors are all linearly independent, and the fourth vector can be linearly expressed by the first three vectors.
Therefore, the rank of a vector group can be obtained by performing an elementary row transformation on it, converting it into row echelon form, and then counting the number of non-zero rows.

Maximally linearly independent group

Maximal Linearly Independent Group is a concept in linear algebra that is used to describe the relationship between vector groups.
The maximal linearly independent group of a vector group is a subset of the vector group that satisfies the following two conditions:
1. The vectors in this subset are linearly independent.
2. Any vector in the vector group can be expressed linearly by the vectors in this subset.
There may be multiple maximal linearly independent groups of a vector group, but the number of vectors they contain is the same, which is called the rank of the vector group.
For example, for the vector group A = ((1, 2, 3), (4, 5, 6), (7, 8, 9)), a maximal linearly independent group of it can be ((1, 2, 3 ), (4, 5, 6)), or ((1, 2, 3), (7, 8, 9)), but their ranks are all 2.
The concept of maximal linearly independent groups is very important in linear algebra because it can be used to determine the independence of vector groups and the dimensionality of generated subspaces.

Rank of the matrix

The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix.
For matrix A, if its row vector group is linearly independent, then its rank is equal to the rank of the row vector group, that is, r(A) = r(a1, a2, ..., am) For matrix A, if its
columns The vector group is linearly independent, then its rank is equal to the rank of the column vector group, that is, the rank of r(A) = r(a1, a2, ..., an) matrix can be transformed
into Row echelon form, and then count the number of non-zero rows to get.
For example, for the matrix A = ((1, 2, 3), (4, 5, 6), (7, 8, 9)), its rank is 3 because it contains three linearly independent vectors.
For the matrix B = ((1, 2, 3), (4, 5, 6), (7, 8, 9), (10, 11, 12)), its rank is also 3, because any three of them The vectors are all linearly independent, and the fourth vector can be expressed linearly from the first three vectors.

Formula for matrix rank

The formula for matrix rank is: R(A)=R(A^T)=R(AA^T)” .

The proof process of this formula is as follows:

First prove that R(A)=R(A^T):

Let B be any r-order subformula of A, and D be any r-order subformula of A^T. According to the properties of the sub-formula, the following two equations can be obtained:

BD=DB=0，

Therefore, D and D^T are both the highest-order non-zero subformulas of A and A^T. The number r is called the rank of the matrices A and A^T, and is recorded as R(A)=R(A^T).

Then prove R(A)=R(AA^T):

Assume C is any r-order subformula of AA^T. According to the properties of AA^T, the following equation can be obtained:

AC=CA=0，

Therefore, A and C are both the highest-order non-zero subformulas of AA^T, and the number r is called the rank of matrix AA^T, denoted as R(AA^T)=r. At the same time, according to R(A)=R(A^T), we can get R(A)=R(AA^T).

Therefore, the formula for matrix rank is R(A)=R(A^T)=R(AA^T)''.

three ranks equal

The "three-rank equality" of matrices means that the ranks of the matrix, the rank of the transpose of the matrix, and the ranks of the product of the matrix and its transpose are equal.

Specifically, assuming $A$ is a $m\times n$ matrix, then there is the following equation:

1. $rank(A) = rank(A^T)$
2. $rank(AA^T) = rank(A)$
3. $rank(A^TA) = rank(A^T)$

Among them, $rank(A)$ represents the rank of matrix $A$, $A^T$ represents the transpose of matrix $A$, and $AA^T$ represents the product of matrix $A$ and its transpose.

This conclusion is very useful in matrix analysis and linear algebra, and it can be used to solve some problems related to matrix rank.