Linked Questions
32 questions linked to/from Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix
1
vote
3
answers
5k
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 ...
- 860
3
votes
1
answer
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?
- 13.9k
0
votes
2
answers
1k
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 \...
- 428
3
votes
1
answer
385
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
0
answers
17
views
Calculate column rank of a matrix [duplicate]
How do I calculate the column rank of the matrix:
$$\begin{bmatrix} -2 & -4 & -6 & 0 \\ 0 & -12 & 24 & 0 \\ 0 & 0 & 28 & -4 \\ 0 & 0 & 0 & 0\end{bmatrix}...
- 147
200
votes
91
answers
57k
views
Which one result in mathematics has surprised you the most? [closed]
A large part of my fascination in mathematics is because of some very surprising results that I have seen there.
I remember one I found very hard to swallow when I first encountered it, was what is ...
103
votes
7
answers
108k
views
Geometric interpretation of $\det(A^T) = \det(A)$
$$\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?
- 3,701
31
votes
1
answer
11k
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 ...
- 11.4k
13
votes
5
answers
12k
views
Why do elementary matrix operations not affect the row space of a given matrix?
I have shown that two of the three elementary operations will not change the image of the row space of the matrix: given a row vector $\vec{v}$, $k\vec{v}$ will span the same (scalar multiplication), ...
- 1,959
8
votes
2
answers
5k
views
Why do row operations 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 prove ...
- 336
12
votes
1
answer
39k
views
Finding null space of matrix.
I need to make sure I'm understanding this correctly.
I skipped a few steps to reduce typing, but let me know if I need to clarify something.
Question asks:
Find $N(A)$ for $A$ = \begin{bmatrix}
...
- 349
2
votes
3
answers
3k
views
Complex conjugation does not change rank
I have a question about complex conjugation of a matrix. Prove that for any rectangular matrix $A$ the following holds
rank $A = \text{rank} \, A^*$ where $A^*$ is complex conjugate transpose of $A$.
- 83
0
votes
2
answers
5k
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 ...
- 11
0
votes
1
answer
4k
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 ...
- 531
4
votes
2
answers
448
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}} (...
- 4,659