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.
Near-dependence of columns can produce a much larger condition number. While the exact rank might be full, the geometric closeness of the columns to a lower-dimensional subspace makes the matrix ill-conditioned, leading to sensitivity in calculations.
Conditions: The columns are linearly independent but nearly dependent.; The matrix has full exact rank.
Near-dependence of columns can produce a much larger condition number. While the exact rank might be full, the geometric closeness of the columns to a lower-dimensional subspace makes the matrix ill-conditioned, leading to sensitivity in calculations.
Conditions: The columns are linearly independent but nearly dependent.; The matrix has full exact rank.