2cm Most people are afraid to admit that they don’t know the answer to some question, and as a consequence they refrain from mentioning the question, even if it is a very natrual one. What a pity! As for myself, I enjoy saying “I do not know”.
– Jean-Pierre Serre (1926 - )
Fields medal and Abel prize winner
If you have two vectors in a vector space, some natural questions come to mind: How do their lengths compare? What is the angle between them? Using just the vector space axioms, these questions are unanswerable. But if you are additionally given an inner product on the vector space, then the questions become answerable. I recommend that throughout you try to form geometric pictures in your mind.
Concepts from this Chapter are used throughout mathematics and statistics. In geometry, orthogonality is used to understand surfaces in (see MATH329), in probability and statistics, covariance is a bilinear form on the space of random variables (see MATH230, and many other modules), in analysis, inner products are used to understand infinite-dimensional vector spaces (see MATH314, MATH317), in combinatorics, orthogonality is used to study Latin squares (see MATH327).