Relation between condition and linear dependence of column vectors There are several interpretations of the condition number of a matrix: Relation between smallest and largest singular values, amount of error amplification etc.
In my opinion, another interpretation should be how far the column vectors (i.e., the image/range space) diverge from being perfectly orthogonal. Perfectly orthogonal column vectors would mean condition number of 1 and linear dependent ones in very high condition numbers.
Is this interpretation correct?
Is there a similar interpretation for the row vectors?
In any case, I tried to verify this idea with simple 2x2 matrices and in most cases, it is indeed true (the closer the normalized dot product between the column vectors to 0 (i.e., 90 degrees), the lower the condition number). But not always.
For example:
$$
A_1 = \begin{pmatrix}
 0 & 4 \\ 15 & -2
\end{pmatrix} ,\quad
A_2 = \begin{pmatrix}
 15 & -2 \\ -8 & 11
\end{pmatrix}
$$
$\mathrm{cond}(A_1) = 3.82$, dot product of column vectors is -0.44, dot product of row vectors -0.13. But $\mathrm{cond}(A_2) = 2.35$, dot product of column vectors is -0.62, dot product of row vectors -0.69.
So $A_2$ has the lower condition number but the vectors seem to be less "orthogonal" than $A_1$.
Why is this the case?
 A: It depends how you measure the orthogonality. Normally, the orthogonality is measured in terms of some "loss of orthogonality", that is, some quantity penalizing the fact that $A$ is not orthogonal (= having orthonormal columns).
One such measure could be to take a norm of $Q-A$, where $Q$ is the orthogonal matrix closest to $A$ (in some norm) such that the column space of $Q$ and $A$ are identical. 
Assume that $A=USV^T$ is the SVD of $A$. For the 2-norm, it can be shown that such a matrix is given by $Q=UV^T$, which is actually the orthogonal factor of the polar decomposition of $A$. To make some condition number appear there, assume that $\|A\|_2=1$ (if we would like to measure the orthogonality of $A$, we could at least scale it to unit norm). Then (using the fact that the 2-norm is orthogonally invariant and the 2-norm of a diagonal matrix is equal to the maximal diagonal entry in the absolute value) we have
$$
\frac{\|Q-A\|_2}{\|A\|_2}=\frac{\|UV^T(I-VSV^T)\|_2}{\|S\|_2}=\frac{\|I-S\|_2}{\|S\|_2}=1-\mathrm{cond}(A).
$$
If $A$ was square, we would get the same quantity for rows of $A$, although, if $A$ was not of full rank, the matrix $Q$ (with the properties mentioned above) would be generally different for rows and columns (but $Q=I$ in the full rank case).
Another useful meaning of $\mathrm{cond}(A)$ is the relative reciprocal distance to the nearest matrix of lower rank, that is,
$$
\frac{1}{\mathrm{cond}(A)}=\min_B\left\{\frac{\|B-A\|_2}{\|A\|_2}:\;\mathrm{rank}(B)<\mathrm{rank}(A)\right\}.
$$
This is closely related to the low-rank approximations.
