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The rank of a matrix being so important, it is frustrating not really knowing what it is, or rather what it represents. The rank of a linear map is the dimension of its range. I don't see what rank of a matrix, which is no function at all, could signify. My textbook, Friedberg's Linear Algebra, gives the following definition:

If $A \in M_{m \times n}(F)$, we define the rank of $A$, denoted $\mathrm{rank}(A$), to be the rank of the linear transformation $L_A : F^n \to F^m$ (left-multiplication transformation)

This of course helps establish some very important results, but I just don't see how the definition 'makes sense' (i.e. compare it to the $\epsilon-\delta$ defintion of a limit which has corresponding intutive sense)

user_hello1
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    Not quite clear what exactly does not make sense. Every matrix $A$ defines a linear map $L(x)=Ax$. The rank of a linear map is the dimension of its range. What are you looking for? – A.Γ. Oct 26 '19 at 10:20
  • Are you ok with the definition of the rank of a linear transformation? Matrices exist in order to represent linear transformations, so it seems natural to define the rank of $A$ to be the rank of a linear transformation that $A$ represents. When I think of a matrix $A$, I think of the mapping $x \mapsto Ax$. – littleO Oct 26 '19 at 10:21
  • If you want a geometric meaning, rank is a number that tells you whether the image $L_A(F^n)$ looks like a point, a line, a plane, something occupying space, or something else. – edm Oct 26 '19 at 10:31
  • Your question is a bit strange. It is as if you said "The protagonist of a story is its main character. I don't see what the protagonist of a book, which is no story at all, could signify". – Ben Grossmann Oct 26 '19 at 11:05
  • Possible duplicate of Importance of matrix rank – Carmeister Oct 26 '19 at 20:42

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Actually rank gives the dimension of column space and row space.It is the number of linearly independent rows or columns of a matrix.It is also the order of the largest square submatrix with non zero determinant.The best way is to think is through system of linear equations,It shows how many equations you actually have i.e.number of independent equations.

Kishalay Sarkar
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Consider a $3\times 4$ matrix $A$

When you multiply $A$ by a column vector $X\in \mathcal {R^4}$ you get a column vector $AX=Y\in \mathcal {R^3}$ thus this matrix defines a linear function $$F:\mathcal {R^4}\to \mathcal{R^3}$$

The range of this function is a subspace of the three dimensional space $\mathcal{R^3}$ so it can be of dimension $1$, $2$, or $3$ which is called the rank of the matrix $A$

Fortunately the rank of matrix is the number of linearly independent rows or columns of that matrix, that is row rank and column rank are the same as the rank of a given matrix.

Mohammad Riazi-Kermani
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If you want a geometric meaning, rank is a number that tells you whether the image $L_A(F^n)$ looks like a point, a line, a plane, something occupying space, or something else.

edm
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