Stillwell's Naive Lie theory was essentially written as an answer to this question. I quote from the introduction:
It seems to have been decided that undergraduate mathematics today rests on two foundations: calculus and linear algebra. These may not be the best foundations for, say, number theory or combinatorics, but they serve quite well for undergraduate analysis and several varieties of undergradu- ate undergraduate algebra and geometry. The really perfect sequel to calculus and linear algebra, however, would be a blend of the two—a two —a subject in which calcu- lus calculus throws light on linear algebra and vice versa. Look no further! This perfect blend of calculus and linear algebra is Lie theory (named to honor the Norwegian mathematician Sophus Lie—pronounced “Lee ”).
