Advanced Engineering Mathematics, Lecture 1.4: Inner products and orthogonality

Advanced Engineering Mathematics, Lecture 1.4: Inner products and orthogonality

In R^n, the length (norm) of vectors and the angles between vectors can all be defined using a dot product. This also gives rise to basic Euclidean geometry, such as the law of cosines and projections. The generalized version of the dot product in an abstract vector space, written (v,w), is called an "inner product", and it endows a vector space with a basic notion of geometry. Two vectors for which (v,w)=0 are said to be "orthogonal", which is the generalized version of perpendicular. We conclude with a few examples of inner products in spaces of periodic functions. In the latter example, the inner product is defined as an integral, and this makes either the complex exponentials, or the sines and cosine waves, into an orthonormal basis. This leads to the notion of a Fourier series, and how to write any periodic function using an infinite sum of complex exponentials, or sines and cosines.

TYPO: Page 4/8, Definition (i), should clearly be (u+v,w)=(u,w)+(v,w), not (u+v,w)=(u,v)+(v,w).

[enesalbay]

@ 15:42 There is a mistake in the fist condition of bilinearity. It should be <u+v, w>=<u, w> + <v, w>, not <u+v, w>=<u, v> + <v, w>.