I too never really grokked some essential points of linear algebra until I (re)learned it from a chapter in Hungerford's Algebra book, part of Springer's yellow GTM series. I was taught from Strang in undergrand, and while I did well in the course, I often found myself blind as to what all these computations meant. I recall that Strang had said in his preface that the subject had been taught too abstractly and the crucial importance missed, which reading Greub e.g. or other treatises of that era may be true.