Strang's book is very bad as an intro to linear algebra, especially for someone to use on his/her own. Topics are obscurely sequenced, and motivation is missing, unless you count Strang's "this family of matrices is terrific" and other comparable expressions as motivation. I would bet that not a single page has gone under the eyes of a competent editor. Some may call the style "conversational", but it isn't, unless you accept statements like "a and b are most likely orthogonal" as conversational; in my opinion they don't belong in a mathematics textbook. For a good intro linear algebra try Nicholson.