BIZNIQUE / LINEAR ALGEBRA / 10
When a transformation loses a dimension
When a transformation loses a dimension
Collapse a plane onto a line and see why an inverse can disappear.
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THE IDEA
Read the example.
Read the example.
For a square matrix, singularity means at least one nonzero vector maps to zero. Distinct inputs can produce the same output, so a unique inverse cannot reconstruct the original input. The example shrinks the second axis towards zero.
det(A) = 0 ⇒ A is singular
Use it well
Use it well
A tiny nonzero determinant alone is not a scale-independent diagnosis of numerical trouble. Conditioning and singular values give a more informative picture.
TRY IT YOURSELF
A quick check.
A quick check.
If A = [[1, 0], [0, 0]], what happens to (2, 3) and (2, −4)?
Check your reasoning
Check your reasoning
Both map to (2, 0). Their difference lies in the nullspace.
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