Is the rank of row ladder matrix equal to the final row simplest matrix? Is the rank determined by row ladder or row simplest

Is the rank of row ladder matrix equal to the final row simplest matrix? Is the rank determined by row ladder or row simplest

equally
Rank is the number of nonzero rows in the simplest form of a ladder matrix or row

If the rank matrix is the simplest form, I can find it again,

There's no need to simplify
To find the rank of matrix (or vector group), maximal independent system and judge the existence of solutions of equations, we only need to change the ladder form
Finding the linear representation, using the maximum independent system to represent the rest of the vectors, finding the general solution of the equations, need to be reduced to the simplest form

Finding the simplest ladder of matrix rank
1 1 -2 2
0 2 -1 3
-1 0 1 1
Detailed steps

r3+r1
1 1 -2 2
0 2 -1 3
0 1 -1 3
r1-r3,r2-2r3
1 0 -1 -1
0 0 1 -3
0 1 -1 3
r1+r2,r3+r2
1 0 0 -4
0 0 1 -3
0 1 0 0
r2r3
1 0 0 -4
0 1 0 0
0 0 1 -3

In linear algebra, how to transform an augmented matrix into a row ladder matrix

The first non-zero element in the first row multiplies the correlation coefficient to change other numbers in the column of the non-zero element into zero elements. Look at the first non-zero element in the second row. By analogy, it becomes a ladder row. If the first row is all zero, then find a row that is not all zero to swap with the first row

Tongji's linear algebra, page 61 said that any matrix can always be transformed into row ladder type matrix through finite elementary row transformation. How to prove?
I know, obviously, you can figure it out when you think about the equations. I'm talking about proof

Letters, slowly,

What is row ladder matrix? What is row simplest matrix?

Define a row ladder matrix. If (1) the first non-zero element of each non-zero row is 1; (2) all other elements in the column of the first non-zero element of each non-zero row are zero, it is called row simplest matrix

Is a matrix obtained by multiplying a matrix by another matrix? Is that a matrix obtained by multiplying a row matrix by a column matrix?

If it can be multiplied, then the matrix multiplied by the matrix will of course get the matrix (here the number is regarded as a special matrix of one row and one column)
The result of row matrix multiplied by column matrix is a number, which is regarded as a special matrix of one row and one column

How can I prove that a matrix of rank 1 can be transformed into a column vector multiplied by a row vector

Let R (a) = 1
Then a ≠ 0
Let I0 of a not all be zero
Note that the row vectors of a are A1, A2,..., am
Since R (a) = 1, then ai0 is a maximal independent group of the row vector group of A
The row vectors of a can be expressed linearly by AI
Let AI = kiai0
Let B = (K1, K2,..., KM) ^ t
Then bai0 = a
That is, a is the product of a column vector and a row vector

Master: in linear algebra, if matrix A and B are congruent, then B and a are congruent? Why? What's the meaning of putting it in geometry or physics

In linear algebra, if a and B are congruent, then B and a are congruent
A: the contract
Transpose of a = t * b * t
Then the transpose of the inverse of B = t * the inverse of a * t
So the contract
Geometric background:
Two congruent matrices are actually the metric matrices of the same bilinear function on different bases
I believe these things can be found in your textbooks

What is the geometric meaning of matrix?
Is it the angular equation of a vector?

The 3 * 3 matrix geometrically represents the second order tensor
The vector belongs to the first order tensor