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.
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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.Can changing representation reveal a solution? →A different representation makes certain relationships directly visible. Compare which calculation each representation makes easier and which information it preserves.
Compare the passages
A determinant of 0 would mean that all of space is squished onto something with 0 volumeIf these three columns
all lie in the same plane, then their combinations
will lie in that same plane.make this determinant zero, meaning the tweaked transformation squishes space into a lower dimension