The determinant | Chapter 6, Essence of linear algebra
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Further reading 3
Why shift a matrix before looking for an eigenvector? →A zero determinant signals collapse into a lower dimension. Applying that test to the shifted matrix explains how a nonzero vector can lie in its null space.Which equations survive a collapse in dimension? →Once a transformation flattens space, reachable outputs lie in its image. A target inside that image and one outside it lead to different answers.Which outputs disappear when a transformation collapses space? →Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.
Caption notes
The determinant measures signed area or volume scaling. Zero signals collapse into a lower dimension, and composing transformations multiplies their scale factors.
Original passages · 4
the factor by which a linear transformation changes any areaA determinant of 0 would mean that all of space is squished onto something with 0 volumewhenever the orientation of space is inverted, the determinant will be negativethe same as the product of the determinants of the original two matrices