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### 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 ...
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### 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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### 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 \...
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### 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 , ...
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### 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 ...
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### 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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### 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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### 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), ...
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### 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 ...
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### 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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### 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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### $im(A)=im(AA^T)$ [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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### 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 ...