Linear Algebra and Its Applications

Linear Algebra and Its Applications by Gilbert strang Revising this textbook has been a special challenge, for a very nice reason. So many people have read this book, and taught from it, and even loved it. The spirit of the book could never change. This text was written to help our teaching of linear algebra keep up with the enormous importance of this subject—which just continues to grow. One step was certainly possible and desirable—to add new problems. Teaching for all these years required hundreds of new exam questions (especially with quizzes going onto the web). I think you will approve of the extended choice of problems. The questions are still a mixture of explain and compute—the two complementary approaches to learning this beautiful subject.

I personally believe that many more people need linear algebra than calculus. Isaac Newton might not agree! But he isn’t teaching mathematics in the 21st century (and maybe he wasn’t a great teacher, but we will give him the benefit of the doubt). Cer- tainly the laws of physics are well expressed by differential equations. Newton needed calculus—quite right. But the scope of science and engineering and management (and life) is now so much wider, and linear algebra has moved into a central place. May I say a little more, because many universities have not yet adjusted the balance toward linear algebra. Working with curved lines and curved surfaces, the first step is always to linearize. Replace the curve by its tangent line, fit the surface by a plane, and the problem becomes linear. The power of this subject comes when you have ten variables, or 1000 variables, instead of two.

You might think I am exaggerating to use the word “beautiful” for a basic course in mathematics. Not at all. This subject begins with two vectors v and w, pointing in different directions. The key step is to take their linear combinations. We multiply to get 3v and 4w, and we add to get the particular combination 3v +4w. That new vector is in the same plane as v and w. When we take all combinations, we are filling in the whole plane. If I draw v and w on this page, their combinations cv + dw fill the page (and beyond), but they don’t go up from the page.