Section 6.2 Orthogonal Sets A set of vectors u1,u2,,up in Rn is called ...

Section 6.2Orthogonal SetsA set of vectors u 1 ,u 2 ,,u p in R n is called an orthogonal setif u i u j 0 whenever i j.EXAMPLE: Is11,11,00an orthogonal set?001Solution: Label the vectors u 1 ,u 2 , and u 3 respectively. Thenu 1 u 2 u 1 u 3 u 2 u 3 Therefore, u 1 ,u 2 ,u 3 is an orthogonal set.1

THEOREM 4Suppose S u 1 ,u 2 ,,u p is an orthogonal set of nonzerovectors in R n and W spanu 1 ,u 2 ,,u p . Then S is a linearlyindependent set and is therefore a basis for W.Partial Proof: Supposec 1 u 1 c 2 u 2 c p u p 0c 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 0c 1 u 1 u 1 c 2 u 2 u 1 c p u p u 1 0c 1 u 1 u 1 0Since u 1 0, u 1 u 1 0 which means c 1 ____.In a similar manner, c 2 ,,c p canbeshowntobyall0. So S is alinearly independent set. An orthogonal basis for a subspace W of R n is a basis for W thatis also an orthogonal set.2

EXAMPLE: Suppose S u 1 ,u 2 ,,u p is an orthogonal basisfor a subspace W of R n and suppose y is in W. Find c 1 ,,c p sothaty 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 1y 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 c 1 yu 1u 1 u 1Similarly, c 2 , c 3 ,, c p THEOREM 5Let u 1 ,u 2 ,,u p be an orthogonal basis for a subspace W ofR n . Then each y in W has a unique representation as a linearcombination of u 1 ,u 2 ,,u p . In fact, iftheny c 1 u 1 c 2 u 2 c p u pc j yu ju j u jj 1,,p3

EXAMPLE: Express y orthogonal basis110374,as a linear combination of the1 01 , 0 .0 1Solution:yu 1u 1 u 1yu 2u 2 u 2yu 3u 3 u 3Hencey _____u 1 ______u 2 ______u 34

Orthogonal ProjectionsFor a nonzero vector u in R n , suppose we want to write y in R n asthe the followingy multiple of u multiple a vector to uy u u 0y u u u0 and yyuuu u (orthogonal projection of y onto u)z y yuuu u (component of y orthogonal to u)5

83EXAMPLE: Let y and u 41distance from y to the line through 0 and u.. Compute theSolution: yyuuu u Distance from y to the line through 0 and u distance from y toy y y 7

Orthonormal SetsA set of vectors u 1 ,u 2 ,,u p in R n is called an orthonormal setif it is an orthogonal set of unit vectors.If W spanu 1 ,u 2 ,,u p , then u 1 ,u 2 ,,u p is an orthonormalbasis for W.8

Recall that v is a unit vector if v v v v T v 1.Suppose U u 1 u 2 u 3 where u 1 ,u 2 ,u 3 is an orthonormalset.Then U T U u 1Tu 2Tu 3Tu 1 u 2 u 3 ItcanbeshownthatUU T I also. So U 1 U T (such a matrix iscalled an orthogonal matrix).9

THEOREM 6 An m n matrix U has orthonormal columns ifand only if U T U I.THEOREM 7 Let U be an m n matrix with orthonormalcolumns, and let x and y be in R n . Thena. Ux xb. Ux Uy x yc. Ux Uy 0 if and only if x y 0.Proof of part b: Ux Uy 10