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Lecture 36: Introduction to Sets of Orthogonal Vectors (Nicholson Section 5.3/Section 8.1)

Alternate Video Access via MyMedia | Video Duration: 53:20
Description: Defined what it means for a set of vectors to be orthogonal. Demonstrated that if P is a matrix whose columns are mutually orthogonal then P^T P = diagonal matrix.
4:30 --- Started with a discussion of change of basis material that wasn’t discussed and what students aren’t responsible for.
5:45 --- recalled the definition of an orthogonal set of vectors. Discussed specific examples.
9:28 --- Why do we care? A) If a set is orthogonal it’s easy to check if it’s linearly independent. B) If a set’s orthogonal it’s easy to figure out whether a specific vector is in the span of the set. C) If a matrix is symmetric then it’s diagonalizable and you can find an orthogonal basis of eigenvectors.
12:30 --- Proved that an orthogonal set of nonzero vectors is linearly independent. 14:45 --- Misspoke and said “orthogonal” instead of “linearly independent”.
21:20 --- If a vector v is in the span of a set of orthogonal vectors, here’s a fast way of finding the linear combination that equals v.
26:30 --- Stated a theorem which is “Given an orthogonal set of vectors, if v is in the span of the orthogonal set then you can immediately write down a linear combination that equals v.
28:30 --- What goes wrong if v isn’t in the span of the orthogonal set of vectors.
31:45 --- What goes wrong if the set of vectors isn’t orthogonal. Gave an algebraic explanation and a geometric explanation of what goes wrong.
39:45 --- Given an orthogonal set, what’s a fast way of checking if a vector is in their span?
45:45 --- Review of wonderful properties of orthogonal sets of vectors. What’s the cost of this “free lunch”?
48:15 --- Defined what it means for a vector to be orthogonal to a subspace S of R^n. Gave a geometric example of S and vectors that are orthogonal to S.