How does near-dependence of columns affect the condition number?
Conditions
- The columns are linearly independent but nearly dependent.
- The matrix has full exact rank.
Reasoning, step by step
- Observe that the columns are nearly confined to a plane or lower-dimensional subspace.
- Recognize that this near-dependence implies a small singular value.
- Conclude that a small singular value results in a large condition number.
- Note that calculations become sensitive due to this large condition number.
Example
Near-dependence can produce a much larger condition number.
Common misconceptions
- Assuming that full exact rank guarantees a small condition number.
- Believing that near-dependence only affects the visual geometry but not numerical stability.
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