The rank of a matrix $A$ can be viewed as the number of linearly independent columns of said matrix. For example $$ A = \begin{bmatrix} 1 & 1 \\ 2 & 2 \end{bmatrix} $$ has rank $r=1$ and the column space spans a line contained in $\mathbb{R}^2$.
Numerically speaking matrix $B$ is "almost" the same as $A$: $$ B = \begin{bmatrix} 1 & 1.01 \\ 2 & 2 \end{bmatrix} $$ However, strictly speaking, matrix B has rank $r=2$, as its columns spans the entire $\mathbb{R}^2$ space. Even though its rank is 2, in an informal way we could say that B has a rank $R$ "almost 1" ($R \approx 1$ or maybe $1<R<2$) due to the fact that its second column is very close to its first column.
Is there some sort of continuous "rank" $R$ (with values that are not necessarily positive integers but are from $\mathbb{R}$) that captures this property?
Another way of restating the question: is there a generalization of the notion of dimension of a vector space that is not necessarily a positive integer? I suspect fractal theory might offer some insight.
The non-zero singular values and associated singular vectors of a matrix $A$ determine the dimensions and orthogonal bases for $R(A)$, $R(A^{T})$, $N(A)$, and $N(A^{T})$. You can set a tolerance below which a singular value should be considered effectively 0 and use that to determine the numerical rank.