Orthogonal Sets PDF

Orthogonal Sets A set of rectors [U ,..... up] in R" is said to be an...

Orthogonal Sets A set of rectors [U ,..... up] in R" is said to be an orthogonal set if each pair of distinct rectors from the set is orthogonal , that is , if Ui. U = 0 Whenever ifj. Example : Show that &U , Un , 433 is an orthogonal set , where = / =I = , us 1 1 0 Solution : u.. uz = 3( 1) + - 1 2. +. = n.. 4y = 3) 2) - + 1) 2) + - 1(z) = 0 12 43. = = 1) z) - + 2( 2)- + 1. (E) = 0 Theorem : If Sequelz ,...., up3 is an orthogonal set of nonzero vectors in R , then S is linearly independent and hence is a basis for the subspace spanned by. S Definition : An orthogonal basis for a subspace Wof RV is a basis for W that is also an orthogonal. set Theorem : Let S , Uz ,....., up3 be an orthogonal basis for a subspace W of RV. For each yeW , the weights in the linear combination y = cue +.... -- + CpUp are given by ci = Y for j , 1 =..., p. Example : The set Gu 42433 , where u = (3) =( = , , us is an orthogonal basis for I Express the rector y-[] as a linear combination of the rectors in. s Solution Compute g u= 11 y Uz = -12 y Uz = 33 - :... n 1. n , = 11 Uz =. = G Uz Uz = : 33/2 y = + + y = n - 24z - 34z  Orthogonal Projections scalar Problem : Given UEIR" , decompose a rector yeR" into sum of two vectors y = Y+ z:multiplea to u z = y - Y - M let Y=. U W Then y- is orthogonal to u if and only if je iff y projwY = 8= (y (4) -. n = y. u - (u). u = yu x(u n)))d -. = Yu Y is called the orthogonal projection of y onto. u z = y-Y is called the component of y orthogonal to. U This determined L spanned by projection is by the subspace. U Sometimes -projy , and called the orthogonal projection of y onto L. ↑ notation = projy = You are Example : Let y = (5) an (4) = Find orthogonal projection of y outo. u Then write y as the sum of two orthogonal rectors. Soln : y. u = (5)(7 = 40 j = y === ( nu = (2) = 20 [i] · e - 5 = - = => (5) (i) = + - (y j) · - = [] (2) · = 0 g i i y) (y - * Find the distance from y to L = Span[i]} > - Ily-yll V+ 2 5 = =  Geometric Interpretation - > orthogonal Consider R2 Span[un = , 42] In Yz - u y U , y -4z yel. Any vector can be written y = u, + Uz un 41 42. 42 Uzz. < - Y projection= of y onto 12 10Y 0 D projection of y onto y say, = Su , Orthonormal Sets A set [u ,....., Up3 is an orthonormal set if it is an orthogonal set of unit rectors. ↓ W is the subspace spanned by such a set , then EU ,....., up3 is an orthonormal Jasis for. W Example : Show that S4 , V , 233 is an orthonormal basis of I where =I Soln 1 Compute V= 0 v v, = : V.. = 1 V. Vz = O vz V. O 1 V Vz. = Vz Vz. = Theorem : An mxn matrix U has orthonormal columns iff UTU = In Theorem : Let U be an mexn matrix with orthonormal columns & let X , yeRR" Then 1) (lUxll 11 x/ = 3) (Ux) ((y). = 0 iff x. y =. 0 2)((x) ((y). = x y  Example : Let U= /* ) and x() Notice that h was orthonormal columns & U Verify that IlUx1 = 1x1 Solution 1IXII :=: Ux = [i] lux == *** An orthogonal square matrix U is invertible matrix U such that U" = UT. > - So both columns and rows are orthonormal vectors & Orthogonal Projections Example : Let Eu ,...., us] be an orthonormal basis for R and let y = Gu +..... +C4 Consider the subspace W= Spanu u2] , and write y as the sum of a vector z, W & and a rector zze Wt. Solution CUTTsSee : Y = -- z , , is " The Orthogonal Decomposition Theorem : Let W be a subspace of MV. Then each yelR" can be written uniquely in the form y = j+z where YeW & zeWt. In fact , if E , 12 ,...., up3 is any orthogonal y 4 basis of W then y y Up & y.. u, + + Up z y - = - =...... , U ,. , U Up-up Sorthogonal prosection of y on to W Example : Let u= (3) , un = /iR] · Observe that ques is an orthogonal as is for W SpanEu 123 Write the of rector in W & rector orthogonal to W. = ,. y as sum a a Soln : Ye = ] yI  A Geometric Interpretation of the Orthogonal Projections The Orthogonal Decomposition Theorem : Let W be a subspace of MV. Then each yelR" can be written uniquely in the form y = j+z where YeW & zeWt. In fact , if E , 12 ,...., up3 is any orthogonal basis of W , then y =Yuu +.....+ Yup Up & E = yy the Up-up I Orthogonal projection of y on to W a X3 Let W = Spanque 123 , Y 18 I I ! z Uz CX2............ youz y = ne + uz = y , + 42 1 041 4 42 :Uz via i invi each summand in the formula is the orthogonal projection of y onto subspace spanned by 54i3 Properties of Orthogonal Projections *** Let En ,....., up3 be an orthogonal basis for W Then. if yeW Spanun = ,...... up3 then projwy = y. The Best Approximation Theorem : Let W be a subspace of M4 let yelR" be any rector in I" , and i let be the orthogonal projection of y outo W. Then i is the closest point in W to : 1(y - yl) < (ly VII - for all veW distinct from Y. So , is the best approximation to y by elements of W.  Example : If u = [] (i) (2) , u = , y= , and W = Spansussonal Then the closest point in Who is =yuu Example The : distance from a point yel" to a subspace W is defined by as the distance from y to the nearest point in W. Find the distance from y to W= Spanquu23 where -( (i)... - (i) Solution : U.. Uz = 5 1. + ( 2) 2 -. + 1(1) = 0 = GU42] Orthogonal basis projuy y % %u y 42 ) (i) =(i) (i) ( = ) + =.. = = , + a = = = R.. u , Uz Mz - = - 3 + 10 + 10 = 15 y.u , 1 10 18 = 27 -i (i) () (i) - - y uz = - - =. - = u= u = 25 + 4 +1 = 30 42 4z. = 1 + 4 + 1 = 6 1(y - yll = 07 3+ 5 = V = 35. Theorem : If 54....... up3 is an orthonormal basis for a subspace W of IR" , then projwy = (y 4)u. , + (y 42)4z + - - - - - + (y up). up - If U = (n , u.... up] then projuy = UuTy for all yel"  The Gram-Schmidt Process Y algorithm for producing an orthogonal/normal basis for any subspace of I". Theorem : Given a basis EX1 ,...., Xp3 for a nonzero subspace W of R" , define V = Xi Xz V- Vz = Xz - Y 1. V, V V = Xy- e · Up = Xp- -... Up - 1 Then [Y ,...., Vp3 is an orthogonal basis for W. In addition , Spandv ,...., v3 Spanxi = + -- , xk) for 1kp. Example : Let W = Spandx xn3 , where x = /] and x = (2) Construct an orthogonal basis Ev r23, for W. xn = - = (2) ) (2) - = Example : Let X= (x % = , and xx = (0 % + W= Span3x , x, x3 = " Construct an orthogonal basis for W. vi = 4 -) Soln V = , X - : = =  /]r 10) Example : Let W = Spandr mu) , where v = = · Find an orthonormal basis for he orthogonal := m ) (ii): == []