Inner Product Spaces

An inner product space (over the real numbers) is a vector space with an operation called an inner product. A typical example of an inner product space of dimension n is the Euclidean space  R^n with the usual dot product as the inner product (for a = (a_1, a_2, ..., a_n) and b = (b_1, b_2, ..., b_n) in R^n, the inner product of a and b is defined to be <a, b>= a_1b_1 + a_2b_2 + ... + a_nb_n). We will concentrate on R^n and generalize the results to a general inner product space.

We will proceed as follows. First, we give a discussion on vectors in R^2 and R^3 and define the dot product and the cross product of two vectors, then we discuss orthogonal complements and orthogonal basis for a subspace of R^n. Next, we consider the orthogonal projection of a vector onto a dimension two subspace. We also introduce the Gram-Schmidt process to produce an orthogonal basis for a subspace using a given basis of that subspace. Using this process we can give a QR factorization of an invertible matrix. Using the idea of the orthogonal projection of a vector onto a subspace, we form the least-squares problem and provide solutions for such problems. We also use the same idea to study the Fourier series.

The topics presented in this course are included in my course "Linear Algebra" posted on this site. So if you have taken, or are going to take, "Linear Algebra", you don't need to take this course.