Section 6.2, 6.3 Orthogonal Sets, Orthogonal Projections
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1 Section Orthogonal Sets Orthogonal Projections Main Ideas in these sections: Orthogonal set = A set of mutually orthogonal vectors. OG LI. Orthogonal Projection of y onto u or onto an OG set {u u p }; geometry computation interpretation. Expressing y in terms of an OG basis. Projection of y onto subspaces W and W. Orthonormal sets Orthogonal matrices A set of vectors {u... u p } in IR n is an orthogonal set if u i u j = whenever i j. Examples - The set {u u u } = is an OG set because u u = u u = u u =. - The set {v v v } = is not an OG set because v is not OG to v or v.
2 OG LI. Theorem 4 Suppose S = {u... u p } is an OG set of nonzero vectors in IR n. Then S is a linearly independent set. Thus if W = span{u... u p } S is a basis for W. Proof that OG LI Examine the condition for linear independence. That is find all combinations of c i s that make a true statement out of: c u + c u + + c p u p =. Question: OG vectors? What s the difference between LI vectors and An orthogonal basis for a subspace W of IR n is a basis for W that is also an orthogonal set.
3 OG Projection of y onto u; onto {u... u p } Background: Let y u be nonzero unequal vectors in IR n. Express y as y = ŷ + z where ŷ is a vector in the same direction as u and z is perpendicular to u. The vector ŷ in the example above is called the orthogonal projection of y onto u and is denoted proj u y. The vector z in the example above is called the component of y orthogonal to u.
4 Exercises 8. For y = and u = find the orthogonal 4 projection of y onto u. Then write y as the sum of a vector in span{u} and a vector orthogonal to u. 5. For y = 5 that s closest to y. and u = find the point in span{u}. For the problem above find the distance from y to span{u} 4
5 4. Let y = 4 u = 6 6 and u = 4 4. (a) Prove that {u u } is an OG basis for IR. (b) Sketch y u u proj u y and proj u y. (c) What is y (proj u y + proj u y)? Explain. Solution: (a) u u = so u u are OG hence LI. Two LI vectors in IR form a basis of IR. (Could also use Thm. 4) (b) (c) y (proj u y + proj u y) = because projection onto all basis vectors captures every part of y. 5
6 Expressing y in terms of an OG basis Theorem 5 Let {u... u p } be an orthogonal basis for a subspace W of IR n. For each y W the unique weights (c c... c p ) in the linear combination are given by y = c u + + c p u p c j = y u j u j u j for j = p. 5. Express y = 4 in the orthogonal basis as a linear combination of the vectors = {v v v } of IR. Solution: y = proj v y + proj v y + proj v y y v y v = v + v + v v v v = ( /)v + (/)v + ()v y v v v v 6
7 Projection of y onto subspaces W and W 6. Given y = and the OG basis {v v v } = 6 of IR. Define the subspace W by W = span{v v }. Find the component of y that lies within W. [Hint: use theorem 5] 7. For the problem above find the distance from y to W. 7
8 Thm 8: The Orthogonal Decomposition Theorem Let W be a subspace of IR n. Then each y IR n can be written uniquely in the form y = ŷ + z where ŷ W and z W. In fact if {u... u p } is any orthogonal basis of W then ŷ = and z = y ŷ. y u u + u u y u u + + u u y u p u p u p u p Remarks: - The vector ŷ is called the orthogonal projection of y onto W and is denoted by proj W y. - ŷ is the closest vector in W to y. (This is actually Theorem 9 p. 98 in simpler language.) - The distance from y to the subspace W is y ŷ 8
9 Exercise 8. Find the closest point to y in span{u u } where y = 4 u = u =. Solution: The closest point to y in span{u u } is the sum of the components of y along u and u. Thus: closest point = ŷ = y u u u u + y u u u u = 9
10 Orthonormal sets Orthogonal matrices Definitions A set {u... u p } is an orthonormal set if it is an orthogonal set and u i = for i =... p. If W is the subspace spanned by such a set then {u... u p } is an orthonormal basis for W. Example Show that / / / is an orthonormal set. / / = {u u } Solution: - u u = so the vectors are OG. Moreover - u = u u = (/) + ( /) + (/) = 4/9 + 4/9 + /9 = and - u = (/ ) + (/ ) + = / + / =. So {u u } is an orthonormal basis for the subspace (a plane) of IR spanned by this set.
11 Observe: Suppose {u u } is an orthonormal set in IR. Let U be the matrix [ u u ]. Then U T U = u T u T = u u. = u T u u T u u T u u T u Theorem 6 An m n matrix U has orthonormal columns if and only if U T U = I n. Example Recall that orthonormal set. Note that / / / / / / / / / / / / / / / = is an.
12 Other theorems about orthogonal matrices: Theorem 7 Let U be an m n matrix with orthonormal columns and let x and y be in IR n. Then a. Ux = x b. (Ux) (Uy) = x y c. (Ux) (Uy) = if and only if x y = Theorem If {u... u p } is an orthonormal basis for a subspace W of IR n then proj W y = (y u )u + + (y u p )u p. Moreover if U = [ u u u p ] then proj W y = UU T y y IR n. Exercises: see homework
13 Summary Orthogonality LI but LI does not imply orthogonality proj u y = ( ) y u u u u = component of y along u. proj W y = ( ) y u u u u + + y u p u up p u p where {u... u p } is an orthogonal basis of W. [warning this isn t true if the u i are merely LI. They must be OG.] In the above expression u i u i = for i =... p if the basis is orthonormal. Nearest point to y in W distance from y to W Orthogonal matrices U T U = I
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