Linked Questions

3
votes
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?
1
vote
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 ...
3
votes
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 , ...
0
votes
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 \...
195
votes
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 ...
64
votes
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!
20
votes
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 ...
5
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
2
votes
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 ...
4
votes
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}} (...
2
votes
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: ...
4
votes
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, ...

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