The Difference Between the Rank of Vector Group and the Rank of Matrix

The Difference Between the Rank of Vector Group and the Rank of Matrix

The rank of vectors refers to the number of vectors in a maximal linearly independent group
The rank of a matrix is divided into row vector group and column vector group. The rank of these two vector groups is called row vector group and column vector group respectively.

An anecdote of a vector group is the number of vectors in a maximal linearly independent group
The rank of a matrix is divided into row vector group and column vector group. The rank of these two vector groups is called row vector group and column vector group respectively.

What is the difference between the rank of all column vectors composed of the rank sum of matrices? Isn't the rank of a matrix equal to the rank of all column vectors? What's the difference between them?

They are equal
The rank of a matrix equals the rank of the row vector group equals the rank of the column vector group

They are equal
The rank of the matrix equals the rank of the row vector group equals the rank of the column vector group

Excuse me, teacher, why "the rank of a matrix is equal to the rank of its column vector group and is equal to the rank of its row vector group "? How to understand the relationship between the rank of matrix and the rank of vector group. Excuse me teacher, why "the rank of the matrix is equal to the rank of its column vector group, is equal to the rank of its row vector group "? How to understand the relationship between the rank of matrix and the rank of vector group.

First of all, in order to help you understand, you have to make clear two definitions: the definition of the rank of the matrix: the existence of K-order sub-formula is not 0, for any K+1 order sub-formula is 0, then k is the rank of the matrix.

First of all, in order to help you understand, you have to make clear two definitions: the definition of the rank of the matrix: the existence of K-order sub-formula is not 0, for any K+1 order sub-formula is 0, then k is the rank of the matrix.The definition of the rank of the vector group: the maximum linear vector group contains the number of vectors,...

What is the relationship between vector group equivalence and matrix equivalence? Must a matrix of equal rank be equivalent?

A Necessary and Sufficient Condition for Equivalence of Homomorphic Matrices is Rank Equivalence
Vector groups must be expressed linearly with each other if and only if R (A)=R (A, B)=R (B)

A Necessary and Sufficient Condition for Equivalence of Homomorphic Matrices is Rank Equivalence
The vector groups must be expressed linearly with each other if and only if R (A)=R (A, B)=R (B)

The Proof of the Rank of Matrix and the Rank of Its Column Vector Group Tongji Fourth Linear Algebra When the rank of a matrix is equal to the rank of a row vector, the procedure is as follows: Certificate: Let A=(a1, a2,.am) R (A)=r Let, and let the r-order subformula Dr be not equal to 0. Then let Dr not equal to 0 know that the column in which Dr is located is linearly independent. Let any r+1 subformula be zero, and let A know that any r+1 column vector is linearly dependent. My question is: How it is obtained that any r+1 column in A is linearly related from the fact that all r+1 order subformulas are zero? I think that the matrix formed by the elements of r+1 order determinant is linear related from the fact that all r+1 order subformulas are zero. , But we can not obtain that the r+1 column in which it is located is linear correlation, that is, the original vector is linear correlation, then after increasing the dimension, it is not necessarily linear correlation.

Row Rank = Column Rank = Rank of Matrix

What is the rank of a vector group? What is the rank of a matrix?

The matrix is divided into rows, and each row is a vector
These row vectors form a row vector group for A
There is also a column vector group
The conclusion is: rank of A = rank of row vector group = rank of column vector group

The matrix is divided into rows, and each row is a vector
These row vectors form a row vector group for A
There is also a column vector group
The conclusion is that rank of A = rank of row vector group = rank of column vector group

The matrix is divided into rows, and each row is a vector
These row vectors form a row vector group of A
There is also a column vector group
The conclusion is that rank of A = rank of row vector group = rank of column vector group