6 6.2 © 2012 Pearson Education, Inc. Orthogonality and Least Squares ORTHOGONAL SETSSlide 6.2- 2 © 2012 Pearson Education, Inc. ORTHOGONAL SETS  A set of vectors {u1,…,up} in is said to be an orthogonal set if each pair of distinct vectors from the set is orthogonal, that is, if u∙v =0 whenever .  Theorem 4: If S = {u1,…,up} is an orthogonal set of nonzero vectors in , then S is linearly independent and hence is a basis for the subspace spanned by S. ijSlide 6.2- 3 © 2012 Pearson Education, Inc. ORTHOGONAL SETS  Proof: If for some scalars c1,…,cp, then because u1 is orthogonal to u2,…,up.  Since u1 is nonzero, is not zero and so .  Similarly, c2,…,cp must be zero. 10c Slide 6.2- 4 © 2012 Pearson Education, Inc. ORTHOGONAL SETS  Thus S is linearly independent.  Definition: An orthogonal basis for a subspace W of is a basis for W that is also an orthogonal set.  Theorem 5: Let {u1,…,up} be an orthogonal basis for a subspace W of . For each y in W, the weights in the linear combinationSlide 6.2- 5 © 2012 Pearson Education, Inc. ORTHOGONAL SETS  Proof: The orthogonality of {u1,…,up} shows that  Since is not zero, the equation above can be solved for c1.  To find cj for , compute and solve for cj. 2, ,jp KSlide 6.2- 6 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  Given a nonzero vector u in , consider the problem of decomposing a vector y in into the sum of two vectors, one a multiple of u and the other orthogonal to u.  We wish to write ----(1) where for some scalar α and z is some vector orthogonal to u. See the following figure. ˆy y zˆyuSlide 6.2- 7 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  Given any scalar α, let , so that (1) is satisfied.  Then is orthogonal to u if an only if  That is, (1) is satisfied with z orthogonal to u if and  The vector is called the orthogonal projection of y onto u, and the vector z is called the component of y orthogonal to u. z y uˆyyˆySlide 6.2- 8 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  If c is any nonzero scalar and if u is replaced by cu in the definition of , then the orthogonal projection of y onto cu is exactly the same as the orthogonal projection of y onto u.  Hence this projection is determined by the subspace L spanned by u (the line through u and 0).  Sometimes is denoted by projLy and is called the orthogonal projection of y onto L.  That is, ----(2) ˆyˆySlide 6.2- 9 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  Example 1: Let and . Find the orthogonal projection of y onto u. Then write y as the sum of two orthogonal vectors, one in Span {u} and one orthogonal to u.  Solution: Compute 7y64u2Slide 6.2- 10 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  The orthogonal projection of y onto u is and the component of y orthogonal to u is  The sum of these two vectors is y. 7 8 1ˆyy6 4 2                  Slide 6.2- 11 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  That is,  The decomposition of y is illustrated in the following figure. 7 8 16 4 2               ˆyyˆ(y y)Slide 6.2- 12 © 2012 Pearson Education, Inc. AN ORTHOGONAL PROJECTION  Note: If the calculations above are correct, then will be an orthogonal set.  As a check, compute  Since the line segment in the figure on the previous slide between y and is perpendicular to L, by construction of , the point identified with is the closest point of L to y. ˆˆ{y,y y}ˆyˆyˆySlide 6.2- 13 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  A set {u1,…,up} is an orthonormal set if it is an orthogonal set of unit vectors.  If W is the subspace spanned by such a set, then {u1,…,up} is an orthonormal basis for W, since the set is automatically linearly independent, by Theorem 4.  The simplest example of an orthonormal set is the standard basis {e1,…,en} for .  Any nonempty subset of {e1,…,en} is orthonormal, too.Slide 6.2- 14 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  Example 2: Show that {v1, v2, v3} is an orthonormal basis of , where  Solution: ComputeSlide 6.2- 15 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  Thus {v1, v2, v3} is an orthogonal set.  Also, which shows that v1, v2, and v3 are unit vectors.  Thus {v1, v2, v3} is an orthonormal set.  Since the set is linearly independent, its three vectors form a basis for . See the figure on the next slide.Slide 6.2- 16 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  When the vectors in an orthogonal set of nonzero vectors are normalized to have unit length, the new vectors will still be orthogonal, and hence the new set will be an orthonormal set.Slide 6.2- 17 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  Theorem 6: An matrix U has orthonormal columns if and only if .  Proof: To simplify notation, we suppose that U has only three columns, each a vector in .  Let and compute ----(3) mnTU U I 1 2 3u u uU  1 1 1 1 2 1 32 1 2 3 2 1 2 2 2 33 3 1 3 2 3 3u u u u u u uu u u u u u u u u uu u u u u u uT T T TT T T T TT T T TUU               Slide 6.2- 18 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  The entries in the matrix at the right are inner products, using transpose notation.  The columns of U are orthogonal if and only if , , ----(4)  The columns of U all have unit length if and only if , , ----(5)  The theorem follows immediately from (3)–(5). 1 2 2 1u u u u 0TT1 3 3 1u u u u 0TT2 3 3 2u u u u 0TT11u u 1T22u u 1T33u u 1TSlide 6.2- 19 © 2012 Pearson Education, Inc. ORTHONORMAL SETS  Theorem 7: Let U be an matrix with orthonormal columns, and let x and y