Abstract Thinking

While linear algebra has many real-life applications, it also has its elegant side, a side which is somewhat abstract. This aspect of the subject also has a real-life application: to develop clarity of thought and expression. At many points in your professional life, you will need to explain to other people what you are doing, and indeed why you are doing it. The "other people " could include those who control the money you need for your project. Succeeding requires good communication skills, and the key to convincing others is being clear about your ideas. One thing you can learn from the definitions, theorems and proofs you'll see in Linear Algebra (and any branch of pure Mathematics for that matter) is how to think clearly and express yourself clearly, to avoid misunderstanding and confusion. You will find, that in learning linear algebra, your practice in sorting out ideas (some of which will seem quite bizarre at first) will help you to think clearly. In practice, that could matter much more than any particular technical skill you acquire.

One advantage linear algebra has over some other subjects for introducing abstract thinking, is that much of the material has a geometric interpretation. In low dimensions, one can "visualize" algebraic results, and happily, the converse is also true: linear algebra helps develop your geometric instinct. The geometric intuition you already have will be complemented by an "algebraic picture", one that will allow you, with practice, to "see" in higher dimensions that are inaccessible to our normal senses.