Giving a column a nonzero component in a previously unused dimension makes the columns linearly independent, thereby increasing the exact rank to full rank. Even if this perturbation is tiny, the exact mathematical rank changes, although the columns remain nearly dependent.
Conditions: The original columns are linearly dependent.; A perturbation adds a nonzero component to a column in a new dimension.
Giving a column a nonzero component in a previously unused dimension makes the columns linearly independent, thereby increasing the exact rank to full rank. Even if this perturbation is tiny, the exact mathematical rank changes, although the columns remain nearly dependent.
Conditions: The original columns are linearly dependent.; A perturbation adds a nonzero component to a column in a new dimension.