If a vector in the set of vectors $$\{v_1,v_2,v_3...v_n\}$$ can be written as a linear combination of other vectors, must a row in the matrix formed by linearly combining the vectors be equal to a linear combination of other rows? Or, in other words, does column linear dependence imply row linear dependence and vice versa?
Since people talk of linear systems being 'linearly dependent', I would think that the answer to my questions would be yes. However, this example makes me think otherwise:
$$\left(\begin{array}{c} 4 & 1 & 5 \\ -2 & 7 & 6 \\ 2 & 8 & 11 \end{array}\right)$$
The third row is equal to the sum of the first two rows, so the rows are linearly dependent. The first vector is not, however, a linear combination of the other vectors. The second vector is not a linear combination of the others. The third vector is not a linear combination of the others. No vector is a scalar multiple of another. Unless those statements are false, the columns must be linearly independent despite the rows being linearly dependent.