Let A be a real matrix of m*n and A^TA be a positive definite matrix. It is proved by data that r (A^TA)≤r (n)≤n, but I don't know where this formula comes from, and the formula is r (AB)≤r (A), r (AB)≤r (B),. I've found the problem in the book, and now there's another question: Shouldn't r (A)≤min {m, n}? Why is the formula directly r (n)≤n? Somebody help me.

Let A be a real matrix of m*n and A^TA be a positive definite matrix. It is proved by data that r (A^TA)≤r (n)≤n, but I don't know where this formula comes from, and the formula is r (AB)≤r (A), r (AB)≤r (B),. I've found the problem in the book, and now there's another question: Shouldn't r (A)≤min {m, n}? Why is the formula directly r (n)≤n? Somebody help me.

1. Because A*A'(' denotes transposed) is a matrix of n*n, and the rank of a matrix must be less than its number of rows or columns, r (A*A')≤n can be obtained directly.
2. It should be noted that what is n in r (n)? You may be mistaken, a number does not have to be rank (a non -0 number has rank 1,0 has rank 0).

Let A be m*n matrix, B be k*n matrix, and r (A)+r (B)

Let the upper block of a block matrix C be A and the lower block B
The solution of Cx=0 is the common solution of Ax=0 and Bx=0
R (C)

Let A be a real matrix of m*n, n

As known, r (A)= r (A, b)= n
And because A is a real matrix, there is r (A' A)=r (A)=n
So A' A is an n-th order invertible matrix.

As known, r (A)= r (A, b)= n
Because A is a real matrix, there is r (A' A)=r (A)=n
So A' A is an n-th order invertible matrix.

Let A be a m×n matrix, and the sufficient condition for a homogeneous system of linear equations Ax=0 to have only zero solution is () Linear independence of column vectors for A.A Column Vector Linear Correlation for B.A LINEAR INDEPENDENCE OF LINEAR VECTORS FOR C.A Linear correlation of row vectors for D.A Let A be a m×n matrix, and the sufficient condition for a homogeneous system of linear equations Ax=0 to have only zero solution is () LINEAR INDEPENDENCE OF COLUMN VECTORS FOR A.A Column Vector Linear Correlation for B.A LINEAR INDEPENDENCE OF LINEAR VECTORS FOR C.A LINEAR CORRELATION OF ROW VECTORS FOR D.A

A is m×n matrix, A has m rows and n columns, and the equations have n unknowns
Ax=0 has only zero solution. The rank of A is not less than the unknown number n of the system of equations.
R (A)= column rank of n A = column vector of n A is linearly independent.
Matrix A has n columns, and the column vector group of A is linearly independent
But A has m rows, m may be less than n, where row vector group is linear independent, only R (A)= R (A)=m, can not prove r (A)≥n
Therefore, A.

Matrix A (m*n), then the necessary and sufficient conditions for a system of linear equations AX=O to have a zero solution are? Select LINEAR INDEPENDENCE OF LINEAR VECTOR GROUP FOR A.A LINEAR INDEPENDENCE OF ROW VECTOR GROUP FOR B.A Linear independence of column vector group of C.A D. None of the above are correct Pick what? Why? Is there any difference between ranks A is linear correlation Matrix A (m*n), then the necessary and sufficient conditions for a system of linear equations AX=O to have a zero solution are? Select LINEAR INDEPENDENCE OF LINEAR VECTOR GROUP FOR A.A LINEAR INDEPENDENCE OF ROW VECTOR GROUP FOR B.A LINEAR INDEPENDENCE OF COLUMN VECTOR GROUP FOR C.A D. None of the above are correct Pick what? Why? Is there any difference between ranks A is linear correlation

You can try to write out the equations ~ the rows of the coefficient matrix A, i.e. represent the number of equations in the equations, the row-linear independence is that there are m equations ~ the number of columns is the number of variables ~ the necessary and sufficient conditions for only zero solution.

You can try to write the system of equations ~ the row of the coefficient matrix A, which represents the number of equations in the system of equations, the row-linear independence is that there are m equations ~ the number of columns for the number of variables ~ the necessary and sufficient conditions for only zero solution.

Let A be a m*n matrix, then the necessary and sufficient condition for a homogeneous linear system AX=0 to have only a nonzero solution is () Linear independence of column vector group of 1A Linear correlation of column vector group of 2A Linear independence of row vector group of 3A Linear Correlation of Row Vector Groups for 4A The answer is D, why? By the way, please explain the other items Let A be a m*n matrix, then the necessary and sufficient condition for a homogeneous linear system AX=0 to have only a nonzero solution is () Linear independence of column vector group of 1A Linear Correlation of Column Vector Groups for 2A Linear independence of row vector group of 3A Linear Correlation of Row Vector Group of 4A The answer is D, why? By the way, please explain the other items Let A be a m*n matrix, then the necessary and sufficient condition for a homogeneous linear system AX=0 to have only a nonzero solution is () Linear independence of column vector group of 1A Linear correlation of column vector group of 2A Linear independence of row vector group of 3A Linear correlation of row vector group of 4A The answer is D, why? By the way, please explain the other items

AX=0 Linear correlation of column vector groups with nonzero solution A
AX =0 Column vector group linearly independent of non-zero solution A only

Should be (B) correct

AX=0 Linear Correlation of Column Vector Groups with Nonzero Solution A
AX =0 Linear independence of column vector group for non-zero solution A only

Should be (B) correct

AX=0 Linear correlation of column vector groups with nonzero solution A
AX =0 Linear independence of column vector group for non-zero solution A only

Should be (B) correct