Section 6.2 Orthogonal Sets A set of vectors u 1 , u 2 , , u p in - PDF document

Section 6.2 Orthogonal Sets A set of vectors  u 1 , u 2 , … , u p  in R n is called an orthogonal set if u i ⋅ u j = 0 whenever i ≠ j . 1 1 0 EXAMPLE: Is an orthogonal set? , , − 1 1 0 0 0 1 Solution: Label the vectors u 1 , u 2 , and u 3 respectively. Then u 1 ⋅ u 2 = u 1 ⋅ u 3 = u 2 ⋅ u 3 = Therefore,  u 1 , u 2 , u 3  is an orthogonal set. 1

THEOREM 4 Suppose S =  u 1 , u 2 , … , u p  is an orthogonal set of nonzero vectors in R n and W = span  u 1 , u 2 , … , u p  . Then S is a linearly independent set and is therefore a basis for W . Partial Proof: Suppose c 1 u 1 + c 2 u 2 + ⋯ + c p u p = 0  c 1 u 1 + c 2 u 2 + ⋯ + c p u p  ⋅ = 0 ⋅  c 1 u 1  ⋅ u 1 +  c 2 u 2  ⋅ u 1 + ⋯ +  c p u p  ⋅ u 1 = 0 c 1  u 1 ⋅ u 1  + c 2  u 2 ⋅ u 1  + ⋯ + c p  u p ⋅ u 1  = 0 c 1  u 1 ⋅ u 1  = 0 Since u 1 ≠ 0 , u 1 ⋅ u 1 > 0 which means c 1 = ____. In a similar manner, c 2 , … , c p can be shown to by all 0 . So S is a linearly independent set. ■ An orthogonal basis for a subspace W of R n is a basis for W that is also an orthogonal set. 2

EXAMPLE: Suppose S =  u 1 , u 2 , … , u p  is an orthogonal basis for a subspace W of R n and suppose y is in W . Find c 1 , … , c p so that y = c 1 u 1 + c 2 u 2 + ⋯ + c p u p . Solution: y ⋅ =  c 1 u 1 + c 2 u 2 + ⋯ + c p u p  ⋅ y ⋅ u 1 =  c 1 u 1 + c 2 u 2 + ⋯ + c p u p  ⋅ u 1 y ⋅ u 1 = c 1  u 1 ⋅ u 1  + c 2  u 2 ⋅ u 1  + ⋯ + c p  u p ⋅ u 1  y ⋅ u 1 = c 1  u 1 ⋅ u 1  y ⋅ u 1 c 1 = u 1 ⋅ u 1 Similarly, c 2 = , c 3 = , … , c p = THEOREM 5 Let  u 1 , u 2 , … , u p  be an orthogonal basis for a subspace W of R n . Then each y in W has a unique representation as a linear combination of u 1 , u 2 , … , u p . In fact, if y = c 1 u 1 + c 2 u 2 + ⋯ + c p u p then y ⋅ u j c j =  j = 1, … , p  u j ⋅ u j 3

3 EXAMPLE: Express y = as a linear combination of the 7 4 orthogonal basis 1 1 0 . , , − 1 1 0 0 0 1 Solution: y ⋅ u 1 y ⋅ u 2 y ⋅ u 3 u 1 ⋅ u 1 = u 2 ⋅ u 2 = u 3 ⋅ u 3 = Hence y = _____ u 1 + ______ u 2 + ______ u 3 4

Orthogonal Projections For a nonzero vector u in R n , suppose we want to write y in R n as the the following y = multiple of u multiple a vector ⊥ to u +  y − α u  ⋅ u = 0 y ⋅ u − α  u ⋅ u  = 0  α = y = y ⋅ u  u ⋅ u u ( orthogonal projection of y onto u ) and z = y − y ⋅ u u ⋅ u u ( component of y orthogonal to u ) 5

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− 8 3 EXAMPLE: Let y = and u = . Compute the 4 1 distance from y to the line through 0 and u . Solution: y = y ⋅ u  u ⋅ u u = Distance from y to the line through 0 and u = distance from  y to y = ‖  y − y ‖ = 7

Orthonormal Sets A set of vectors  u 1 , u 2 , … , u p  in R n is called an orthonormal set if it is an orthogonal set of unit vectors. If W = span  u 1 , u 2 , … , u p  , then  u 1 , u 2 , … , u p  is an orthonormal basis for W . 8

v T v = 1 . Recall that v is a unit vector if ‖ v ‖ = v ⋅ v = Suppose U = u 1 u 2 u 3 where  u 1 , u 2 , u 3  is an orthonormal set. T u 1 Then U T U = T u 1 u 2 u 3 u 2 = T u 3 = It can be shown that UU T = I also. So U − 1 = U T (such a matrix is called an orthogonal matrix ). 9

THEOREM 6 An m × n matrix U has orthonormal columns if and only if U T U = I . THEOREM 7 Let U be an m × n matrix with orthonormal columns, and let x and y be in R n . Then a. ‖ U x ‖ = ‖ x ‖ b.  U x  ⋅  U y  = x ⋅ y c.  U x  ⋅  U y  = 0 if and only if x ⋅ y = 0 . Proof of part b:  U x  ⋅  U y  = 10