Advanced Linear Algebra - Advanced Linear Algebra, Lecture 2.2: Applications of the rank-nullity theorem

Advanced Linear Algebra, Lecture 2.2: Applications of the rank-nullity theorem Though we could have proven the rank-nullity theorem in the context of matrices and systems of equations, establishing it a general setting of vector spaces leads to far-reaching consequences in a number of diverse areas of mathematics. In this lecture, we will explore some of these. We will begin by showing how any degree-n polynomial can be interpolated uniquely from n+1 input-output pairs. Then, we show how such a polynomial is uniquely determined by its average value on n+1 intervals. Next, we turn to differential equations, and show why the celebrated method of undetermined coefficients works in a special case, which is really not much different than the general case. Finally, we look at the finite difference numerical method of solving Laplace's partial differential equation on a bounded region. We show how the rank-nullity theorem implies that there will always be a unique solution, that is determined by the boundary conditions. Course webpage: http://www.math.clemson.edu/~macaule/math8530-online.html