Linked Questions

1answer
4k views

Intuitive proof of row rank = column rank? [duplicate]

Is it possible to give an intuitive/elementary proof of the theorem that says that the row rank of a (finite-dimensional) square matrix matrix equals its column rank?
3answers
2k views

row rank= column rank, alternative proof [duplicate]

I am studying the theorem that states that the row rank of a matrix is the same as the column rank. I understood the proof and managed to use it in specific examples using a matrix. I am now, trying ...
1answer
326 views

Show that for any matrix $A_{m \times n}$ , the row rank and column rank are equal [duplicate]

Can somebody first please tell me what is the row rank and column rank of a matrix ? What is the relation of each with the rank of a matrix ? Any kind of explanatory proof would be very helpful , ...
2answers
164 views

How come dim row A = rank if dim Im A is also = rank? [duplicate]

The following identities are true for a matrix $A$. \begin{align} \dim \mathrm{row}\, A &= \mathrm{rank}\,A \\ \dim \mathrm{Im}\, A &= \mathrm{rank}\, A \\ \dim \mathrm{row}\, A &= \dim \...
91answers
51k views

Which one result in mathematics has surprised you the most? [closed]

A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is ...
6answers
78k views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
1answer
8k views

Proof that determinant rank equals row/column rank

Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square ...
2answers
2k views

Why Row operation does not change the column rank?

From this question link, I got to know that row operation(row subtraction and row permutation) do change column space. But still it seems that it does not change the column rank. I am trying to ...
2answers
3k views

Rank of vectors

Prove that the rank of a system of vectors from $E^n$ does is not bigger than the dimension of the vectors. For example the vectors $a,b,c$ are from $E^n$ so each of them has $n$ components (the ...
1answer
2k views

$im(A)=im(AA^T)$ [duplicate]

If $A$ is an $n\times m$ matrix, is the formula $im(A)=im(AA^T)$ necessarily true? Explain. I believe this to be true when $n=m$ but am unable to prove if it's true where $n\neq m$. I also don't ...
1answer
498 views

Row & Column Operation to Determine Rank

While evaluating the rank of a matrix is it permissible to apply row and column operations simultaneously on a single matrix? Most of the books that I discussed use either row or column operation (but ...
1answer
572 views

Basis of Column Space of matrix that is in Row Reduced Echelon form

I'm considerably new to proving things(and to Linear Algebra too).So, I hope someone would help me with this. While proving Row Rank of Matrix = Column Rank of Matrix Proof used the point that ...
2answers
104 views

Intuition: If the columns of a matrix are colinear, then its rows are also colinear.

For simplicity's sake, I'm working with a 3x3 square matrix in which none of the column or row vectors is the zero vector. I tried graphing the columns of the matrix {{1,4,-3},{2,7,-5},{3,6,-3}} (...
1answer
195 views

$\dim RS(A) = \dim CS(A)$ from $\dim W^\perp + \dim W = dim V$

Please help me with this: From the formula $\dim W^\perp + \dim W = \dim V$ for general subspaces $W \subset V$ of an inner product space, deduce that the row rank of $A$ is equal to its column rank: ...
0answers
149 views

Unexpected applications of row rank = column rank

The fact that the row rank of a matrix is the same as the column rank is quite surprising (to me atleast, hopefully to you too!). I am looking for unexpected applications of this fundamental fact, ...

15 30 50 per page