serge lang's algebra, advanced as it is overall, may actually have some useful discussions about topics in linear algebra.
lang writes extremely well locally at times. I.e. on a single page, devoted to a single idea, he sometimes has the most succint and clear explanation you can find. then the next sentence may make no sense at all.
i would guess your teacher actually meant to recommend one of langs linear algebra books, some of which are extremely clear, and well organized, at least the edition that i saw years ago. of course his books appear repeatedly in many revised forms, and as students get weaker he watered them down more to accommodate.
but lang's book had sections entitled: "the linear map associated to a matrix". and then: "the matrix associated to a linear map".
these are the key constructions, and they are not always that clearly laid out everywhere.
for example people rave about strang and his "4 fundamental subspaces" but i am not so enchanted with them or him myself.
indeed there are really only two subspaces, kernel and image, but there are two maps associated to a matrix, one to the original matrix and one to its transpose, so you get 4 subspaces.
there are two ways to choose a book, either it is has the best discussion by the most expert author, or it describes the material in a way that speaks to you the reader. a learner often needs both kinds, but no one else can recommend the second kind. you must go to the library and compare books.
the purpose of the book that speaks to you, is to unable you eventually to read the one written by the expert.
fraleigh is an ordinary author of ordinary books, whose (expensive) books are liked by some people, lang is an expert author, but one who writes book at many levels.
if you really want a good book, try bourbaki on linear algebra. it is even translated into english, and all bourbaki books include a historical discussion, and a good one, not just the little thumbnail bio with a funny looking picture of Newton or leibniz.
bourbaki books are fantastically clear especially in linear algebra and commutative algebra, having been written apparently by the world's best experts in the 60's on algebraic geometry.
for undergraduate linear algebra, i like the book of adams and shifrin, for its mingling of geometry with linear algebra, although the emphasis is actually strongly on matrices. the problems are excellent as in all of shifrin's books.
charles curtis' book was recommended to me by a student and looked pretty good as i recall.
but the ebook of sharipov is really quite well done, with a carefully graduated treatment of the essential idea (nilpotency) underlying the more difficult decomposition theorems, and beginning with some elementary material on sets and functions if i recall correctly.
if you like hardbound books, and have a finite budget, that is another reason to look at lang's book, as it is about 1/2 the price of the more popular ones, like strang and shifrin - adams.
notice (sighh...) that sheldon axler's "linear algebra done right" sells for about 1/3 the price of books on linear algebra done wrong.
there are also some good reprinted paperbacks that sell for around 7 - 10 dollars.
e.g. shilov $15, walter nef $8. how can you go wrong at these prices? and these are good books.
these cheap old reprints are 1960's books which were not written to compensate for the weaknesses of todays students, now their only "shortcoming".
oh, and the absolute classic american linear algebra book, done right, is the one by hoffman and kunze, hands down the best. this is probably the only linear algebra book (except bourbaki), that i would bother to have on my own math book shelf, as opposed to my textbook shelf.
another excellent book, essentially unfindable, is the SMSG paperback on linear algebra written for high schools when the "new math" movement was attempting to recast high school math instruction at a higher level.
this whole effort, although excellently represented by great textbooks, was sidetracked by a lack of funding for teacher preparation, and by a reluctance of the program's architects to infringe on the marketplace by actually selling large numbers of the terrific books they produced.
the idea was to energize the american book marketplace and inspire other authors and publishers to produce good texts. this did not happen, and the "new math" books produced by the american marketplace were execrable, and togetehr with the inadequate instruction from unprepared teachers, helped give the movement a very bad name. the original works from SMSG via Yale, are still outstanding but available only on library shelves of schools of education.
summary: the really good books are by bourbaki and hoffman - kunze.
excellent cheap books are by sharipov, shilov, nef.
good traditional texts are by lang and adams - shifrin, at $50-$100 or more.
there is no way i would pay anywhere near $100 for fraleigh myself.
a hint: rather than focusing on reducing matrices accurately, learn what the maps are doing. I.e. learn to understand the part your calculator cannot do for you. and learn the proofs.
His lecture notes are filled with errors and he also recommended "Algebra" by Serge Lang for an introductory course in Linear Algebra, which seems a bit extreme.. 
