5.1 Orthogonal Projections. Orthogonality Recall: Two vectors are said to be orthogonal if their dot product = 0 (Some textbooks state this as a T b =

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Presentation transcript:

5.1 Orthogonal Projections

Orthogonality Recall: Two vectors are said to be orthogonal if their dot product = 0 (Some textbooks state this as a T b = 0) ____ The length (magnitude) of a vector is √v·v A vector is called a unit vector if its length is 1

Orthogonal vector spaces Two vector spaces are said to be orthogonal if every vector in one space is orthogonal to every vector in the other space. Imagine that the front wall of the room is a vector space and the floor of the classroom is also a vector space. Where would the origin have to be? Would those two vector spaces be orthogonal?

Find the dot product of the given vectors on a TI 89 Calculator u = (4,10) v = (-2,3) u = (1,5,7) v = (-1,3,4) Note: To find a dot product on the TI89 Press 2 nd 5 (math) – 4 matrix – L Vector ops – 3 dotP dotP([1,5,7],[-1,3,4])

Orthonormal Vectors The vectors u 1,u 2,u 3 in R n are called orthonormal if they are all unit vectors and orthogonal to each other For two orthonormal vectors u i ·u j = { 1 if i=j 0 if i≠j This implies that if a matrix A comprised of orthonormal vectors multiplied by its transpose can only be the resulting matrix A A T or A T A can only have zeros and ones as its elements.

Problem 2 Find the length of the vector

Problem 2 Solution

Review: Properties of a dot product

Problem 4 Find the angle between the two vector

Solution to problem 4 Use: Find the angle between the vectors:

Problem 10 For which values of k are the vectors perpendicular?

Solution to problem 10

Projections using dot products Note: if v is a unit vector then the formula just becomes (u·v) v To project onto a subspace (rather than an individual vector) the formula becomes: (from MV Calculus)

Problem 28

Solution to Problem 28

Homework p odd, all, all In earlier times, they had no statistics, and so they had to fall back on lies. -STEPHEN LEACOCK Well known facts in statistics: 97.3% of all statistics are just made up % of statistics claim a level of accuracy not justified by method used in achieving the result.