Can you sort of think about
9 vectors in 9 dimensional space
and take their combinations?
That's really the central thought that
you get kind of used to in linear algebra.
Even though you can't really visualize it,
you sort of think you can after a while.
Those 9 columns and all their combinations
may very well fill out the whole 9 dimensional
But if the 9th column happened to be the same as the 8th column and gave nothing new, then probably what it would fill out would be I hesitate even to say this it would be a sort of plane, an 8 dimensional plane inside 9 dimensional space. And it's those 8 dimensional planes inside 9 dimensional space that we have to work with eventually
Equal-area pixelization of the sphere
(with constant Jacobian)
QR factorization using Householder matrices
Borowski & Borwein