It may also be that, for applications (whether within or without mathematics proper) the label "linear algebra" is misleading. I'd think that future physicists might care more about (positive-attitude) "functional analysis" than what the traditional mathematics enterprise calls "linear algebra". Depends.
There is also the conflict, alluded-to in comments and answers, that the orthodox/traditional math texts "advancing" linear algebra are a bit aggressive in demanding compliance-in-orthodoxy from the reader. Contemporary math-phys or phys students may bridle at this, and reasonably so, considering that it is mostly due to the typical curricular/textbook inertia.
E.g., the earlier remark that people are abjured by (presumably orthodox) authors/authorities to "forget about finite-dimensional linear algebra to address infinite-dimensional" disserves everyone (even if it can be interpreted as being "true" in some way). That is, while the naive-est extrapolation of finite-dimensional linear algebra proves inadequate, it is nevertheless the model for what everyone (if they were candid) would like to be true. Thus, the task partly becomes appraisal of how far from this ideal we find ourselves in any particular real-life situation.
The main "life-lesson" is about "continuous spectrum", which is exemplified by Fourier transform and Fourier inversion on the real line. On one hand, "everything is fine", but, when juxtaposing to (perhaps naive) linear algebra, there are problems. But, in fact, it truly is fine, when one adjusts one's notion of the range of happy, useful, extensible answers.
Peter Lax's functional analysis book is certainly vastly more sophisticated than upper-division linear algebra books, but th'guy is very practical-minded, and is not adversarial as a writer. His interests in PDE are physical.
My own notes about functional analysis and linear algebra, at my web page, while not pretentious or anti-physical, are written from a viewpoint which may not be the most useful to the questioner, though, who knows? At least I tried to be honest. :)