Linked Questions

1 vote
3 answers

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
1 answer

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?
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0 votes
2 answers

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 \...
3 votes
1 answer

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

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}...
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200 votes
91 answers

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

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?
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31 votes
1 answer

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 ...
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13 votes
5 answers

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), ...
8 votes
2 answers

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 ...
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12 votes
1 answer

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} ...
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2 votes
3 answers

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 $A$.
0 votes
2 answers

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 ...
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0 votes
1 answer

$im(A)=im(AA^T)$ [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 ...
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4 votes
2 answers

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}} (...

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