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Inner product spaces

The document discusses inner product spaces and orthonormal bases. It defines an inner product space as a vector space with an inner product defined on it. An orthonormal basis is introduced as a set of orthogonal unit vectors that form a basis. The Gram-Schmidt process is presented as a method for transforming a basis into an orthonormal basis. Properties of inner products, such as the Cauchy-Schwarz inequality and orthogonal projections, are covered.

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Inner Product Space
11 - 2 Today ◼ Inner Product Spaces ◼ Orthonormal Bases:Gram-Schmidt Process ◼ Mathematical Models and Least Square Analysis ◼ Applications
11 - 3 Inner Product Spaces (1) (2) (3) (4) and if and only if 〉〈〉〈 uvvu ,, = 〉〈〉〈〉〈 wuvuwvu ,,, +=+ 〉〈〉〈 vuvu ,, cc = 0, 〉〈 vv 0, =〉〈 vv 0=v ◼ Inner product: Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms.
11 - 4 ◼ Note: V Rn spacefor vectorproductinnergeneral, )forproductinnerEuclidean(productdot = = vu vu ◼ Note: A vector space V with an inner product is called an inner product space. ( )•+,,VVector space: Inner product space: ( )•+ ,,,,V
11 - 5 ◼ Ex 1: (The Euclidean inner product for Rn) Show that the dot product in Rn satisfies the four axioms of an inner product. nnvuvuvu +++== 2211, vuvu 〉〈 ),,,(,),,,( 2121 nn vvvuuu  == vu Sol: By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.
11 - 6 ◼ Ex 2: (A different inner product for Rn) Show that the function defines an inner product on R2, where and . 2211 2, vuvu +=〉〈 vu ),(),( 2121 vvuu == vu Sol: 〉〈〉〈 uvvu ,22,)( 22112211 =+=+= uvuvvuvua 〉〈〉〈 〉〈 wuvu wvu ,, )2()2( 22 )(2)(, 22112211 22221111 222111 += +++= +++= +++=+ wuwuvuvu wuvuwuvu wvuwvu ),()( 21 wwb =w
11 - 7 ◼ Note: (An inner product on Rn) 0,, 222111 +++= innn cvucvucvuc 〉〈 vu 〉〈〉〈 vuvu ,)(2)()2(,)( 22112211 cvcuvcuvuvuccc =+=+= 02,)( 2 2 2 1 += vvd 〉〈 vv )0(0020, 21 2 2 2 1 ====+= vvv vvvv〉〈
11- 9 ◼ Ex 3: (A function that is not an inner product) Show that the following function is not an inner product on R3. 332211 2 vuvuvu +−= 〉〈 vu Sol: Let )1,2,1(=v 06)1)(1()2)(2(2)1)(1(,Then −=+−= vv Axiom 4 is not satisfied. Thus this function is not an inner product on R3.
11- 10 ◼ Thm: (Properties of inner products) Let u, v, and w be vectors in an inner product space V, and let c be any real number. (1) (2) (3) 0,, == 〉〈〉〈 0vv0 〉〈〉〈〉〈 wvwuwvu ,,, +=+ 〉〈〉〈 vuvu ,, cc = ◼ Norm (length) of u: 〉〈 uuu ,|||| = 〉〈 uuu ,|||| 2 = ◼ Note:
11 - 11 u and v are orthogonal if . ◼ Distance between u and v: −−=−= vuvuvuvu ,||||),(d ◼ Angle between two nonzero vectors u and v:  = 0, |||||||| , cos vu vu 〉〈 ◼ Orthogonal: 0, =〉〈 vu )( vu ⊥
11- 12 ◼ Notes: (1) If , then v is called a unit vector. (2) 1|||| =v 0 1   v v ⎯⎯⎯ →⎯ gNormalizin v v (the unit vector in the direction of v) not a unit vector
11 - 13 ◼ Ex 6: (Finding inner product) )(inspolynomialbe24)(,21)(Let 2 22 xPxxxqxxp +−=−= nnbababaqp +++= 1100〉,〈 is an inner product ?,)( =〉〈 qpa ?||||)( =qb ?),()( =qpdc Sol: 2)1)(2()2)(0()4)(1(,)( =−+−+=〉〈 qpa 211)2(4,||||)( 222 =+−+== 〉〈 qqqb 22)3(2)3( ,||||),( 323)( 222 2 =−++−= −−=−= −+−=− qpqpqpqpd xxqpc 
11 - 15 ◼ Properties of norm: (1) (2) if and only if (3) ◼ Properties of distance: (1) (2) if and only if (3) 0|||| u 0|||| =u 0u = |||||||||| uu cc = 0),( vud 0),( =vud vu = ),(),( uvvu dd =
11 - 16 ◼ Thm： Let u and v be vectors in an inner product space V. (1) Cauchy-Schwarz inequality: (2) Triangle inequality: (3) Pythagorean theorem : u and v are orthogonal if and only if |||||||||||| vuvu ++ Theorem 5.5 222 |||||||||||| vuvu +=+ Theorem 5.6 |||||||||,| vuvu 〉〈 Theorem 5.4
11 - 17 ◼ Orthogonal projections in inner product spaces: Let u and v be two vectors in an inner product space V, such that . Then the orthogonal projection of u onto v is given by 0v  v vv vu uv   = , , proj ◼ Note: If v is a init vector, then . The formula for the orthogonal projection of u onto v takes the following simpler form. 1||||, 2 == vvv 〉〈 vvuuv = ,proj
11 - 18 ◼ Ex 10: (Finding an orthogonal projection in R3) Use the Euclidean inner product in R3 to find the orthogonal projection of u=(6, 2, 4) onto v=(1, 2, 0). Sol: 10)0)(4()2)(2()1)(6(, =++= vu 5021, 222 =++= vv )0,4,2()0,2,1(proj 5 10 ==   = v vv vu uv
11 - 19 ◼ Thm: (Orthogonal projection and distance) Let u and v be two vectors in an inner product space V, such that . Then0v     vv vu vuuu v , , ,),()proj,( ccdd
11 - 20 Keywords ◼ inner product: 內積 ◼ inner product space: 內積空間 ◼ norm: 範數 ◼ distance: 距離 ◼ angle: 夾角 ◼ orthogonal: 正交 ◼ unit vector: 單位向量 ◼ normalizing: 單範化 ◼ Cauchy – Schwarz inequality: 科西 - 舒瓦茲不等式 ◼ triangle inequality: 三角不等式 ◼ Pythagorean theorem: 畢氏定理 ◼ orthogonal projection: 正交投影
11 - 21 Today ◼ Length and Dot Product in Rn(Continued) ◼ Inner Product Spaces ◼ Orthonormal Bases:Gram-Schmidt Process ◼ Mathematical Models and Least Square Analysis ◼ Applications
11 - 22 Orthonormal Bases: Gram-Schmidt Process ◼ Orthogonal: A set S of vectors in an inner product space V is called an orthogonal set if every pair of vectors in the set is orthogonal. ◼ Orthonormal: An orthogonal set in which each vector is a unit vector is called orthonormal.       = = = ji ji VS ji n 0 1 , ,,, 21 vv vvv   0, ,,, 21 = = ji n VS vv vvv  ji  ◼ Note: If S is a basis, then it is called an orthogonal basis or an orthonormal basis.
5 - 23 ◼ Ex 1: (A nonstandard orthonormal basis for R3) Show that the following set is an orthonormal basis.          −        −          = 3 1 , 3 2 , 3 2 , 3 22 , 6 2 , 6 2 ,0, 2 1 , 2 1 321 S vvv Sol: Show that the three vectors are mutually orthogonal. 0 9 22 9 2 9 2 00 23 2 23 2 00 32 31 6 1 6 1 21 =+−−= =+−= =++−= vv vv vv
11 - 24 Show that each vector is of length 1. Thus S is an orthonormal set. 1|||| 1|||| 10|||| 9 1 9 4 9 4 333 9 8 36 2 36 2 222 2 1 2 1 111 =++== =++== =++== vvv vvv vvv
11 - 25 The standard basis is orthonormal. ◼ Ex 2: (An orthonormal basis for ) In , with the inner product )(3 xP 221100, bababaqp ++= },,1{ 2 xxB = )(3 xP Sol: ,001 2 1 xx ++=v ,00 2 2 xx ++=v ,00 2 3 xx ++=v 0)1)(0()0)(1()0)(0(, ,0)1)(0()0)(0()0)(1(, ,0)0)(0()1)(0()0)(1(, 32 31 21 =++= =++= =++= vv vv vv Then
11 - 26 ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1110000 ,1001100 ,1000011 333 222 111 =++== =++== =++== v,vv v,vv v,vv
11 - 27 ◼ Thm : (Orthogonal sets are linearly independent) If is an orthogonal set of nonzero vectors in an inner product space V, then S is linearly independent.  nS v,,v,v 21= Pf: S is an orthogonal set of nonzero vectors 0and0i.e. = iiji ji v,vv,v iccc ccc iinn nn ==+++ =+++ 0,0, 0Let 2211 2211 vvvvv vvv   = +++++ iii inniiiii c cccc v,v v,vv,vv,vv,v 2211  t.independenlinearlyis00 Siciii = v,v
11 - 28 ◼ Corollary to Thm: If V is an inner product space of dimension n, then any orthogonal set of n nonzero vectors is a basis for V.
11 - 29 ◼ Ex 4: (Using orthogonality to test for a basis) Show that the following set is a basis for .4 R )}1,1,2,1(,)1,2,0,1(,)1,0,0,1(,)2,2,3,2{( 4321 −−−−=S vvvv Sol: : nonzero vectors 02262 02402 02002 41 31 21 =−−+−= =−++−= =−++= vv vv vv 4321 ,,, vvvv 01201 01001 01001 43 42 32 =+−+= =+++−= =+++−= vv vv vv .orthogonalisS 4 forbasisais RS (by Corollary to Theorem 5.10)
11 - 30 ◼ Thm: (Coordinates relative to an orthonormal basis) If is an orthonormal basis for an inner product space V, then the coordinate representation of a vector w with respect to B is },,,{ 21 nB vvv = },,,{ 21 nB vvv  = is orthonormal ji ji ji  =    = 0 1 , vv Vw nnkkk vvvw +++= 2211 (unique representation) Pf: is a basis for V},,,{ 21 nB vvv = nn vvwvvwvvww +++= ,,, 2211 
11 - 31 ik kkk kkk i inniiii inni = ++++= +++= 〉,〈〉,〈〉,〈 〉,)(〈〉,〈 11 2211 vvvvvv vvvvvw   nn vvwvvwvvww +++= ,,, 2211  ◼ Note: If is an orthonormal basis for V and ,},,,{ 21 nB vvv = Vw Then the corresponding coordinate matrix of w relative to B is                  = n B vw vw vw w , , , 2 1 
11 - 32 ◼ Ex 5: (Representing vectors relative to an orthonormal basis) Find the coordinates of w = (5, -5, 2) relative to the following orthonormal basis for . )}1,0,0(,)0,,(,)0,,{( 5 3 5 4 5 4 5 3 −=B 3 R Sol: 2)1,0,0()2,5,5(, 7)0,,()2,5,5(, 1)0,,()2,5,5(, 33 5 3 5 4 22 5 4 5 3 11 =−== −=−−== −=−== vwvw vwvw vwvw           − − = 2 7 1 ][ Bw
11 - 33 ◼ Gram-Schmidt orthonormalization process: is a basis for an inner product space V},,,{ 21 nB uuu = 11Let uv = })({1 1vw span= }),({2 21 vvw span= },,,{' 21 nB vvv = },,,{'' 2 2 n n B v v v v v v 1 1 = is an orthogonal basis. is an orthonormal basis.  − = −=−= − 1 1 〉〈 〉〈 proj 1 n i i ii in nnnn n v v,v v,v uuuv W  2 22 23 1 11 13 3333 〉〈 〉〈 〉〈 〉〈 proj 2 v v,v v,u v v,v v,u uuuv W −−=−= 1 11 12 2222 〉〈 〉〈 proj 1 v v,v v,u uuuv W −=−=
11 - 34 Sol: )0,1,1(11 == uv )2,0,0()0, 2 1 , 2 1 ( 2/1 2/1 )0,1,1( 2 1 )2,1,0( 2 22 23 1 11 13 33 =−−−=   −   −= v vv vu v vv vu uv ◼ Ex 7: (Applying the Gram-Schmidt orthonormalization process) Apply the Gram-Schmidt process to the following basis. )}2,1,0(,)0,2,1(,)0,1,1{( 321 =B uuu )0, 2 1 , 2 1 ()0,1,1( 2 3 )0,2,1(1 11 12 22 −=−=   −= v vv vu uv
11 - 35 }2)0,(0,0),, 2 1 , 2 1 (0),1,(1,{},,{' 321 − == vvvB Orthogonal basis }1)0,(0,0),, 2 1 , 2 1 (0),, 2 1 , 2 1 ({},,{'' 3 3 2 2 − == v v v v v v 1 1 B Orthonormal basis
11 - 36 ◼ Ex 10: (Alternative form of Gram-Schmidt orthonormalization process) Find an orthonormal basis for the solution space of the homogeneous system of linear equations. 0622 07 4321 421 =+++ =++ xxxx xxx Sol:       − − ⎯⎯ →⎯      08210 01201 06212 07011 .. EJG             − +            − =             − +− =              1 0 8 1 0 1 2 2 82 2 4 3 2 1 ts t s ts ts x x x x
11 - 37 Thus one basis for the solution space is )}1,0,8,1(,)0,1,2,2{(},{ 21 −−== vvB ( ) ( ) ( ) ( )1,2,4,3 01,2,,2 9 18 10,8,1, , , 01,2,,2 1 11 12 22 11 −−= − − −−=   −= −== v vv vu uv uv ( )( ) 1,2,4,30,1,2,2' −−−= B (orthogonal basis)             −−       − = 30 1 , 30 2 , 30 4 , 30 3 ,0, 3 1 , 3 2 , 3 2 ''B (orthonormal basis)
11 - 38 Keywords: ◼ orthogonal set: 正交集合 ◼ orthonormal set: 單範正交集合 ◼ orthogonal basis: 正交基底 ◼ orthonormal basis: 單範正交基底 ◼ linear independent: 線性獨立 ◼ Gram-Schmidt Process: Gram-Schmidt過程
End of Chapter 11 - 39

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Introduction to inner product spaces, defining inner products with axioms, examples, and properties including norms, distances, and orthogonality.

Introduction to inner product spaces, defining inner products with axioms, examples, and properties including norms, distances, and orthogonality.

Discussion of fundamental properties like Cauchy-Schwarz inequality, triangle inequality, distances, norms, and orthogonal projections in inner product spaces.

Discussion of fundamental properties like Cauchy-Schwarz inequality, triangle inequality, distances, norms, and orthogonal projections in inner product spaces.

Concepts of orthonormal sets and the Gram-Schmidt process for constructing orthonormal bases, including exercises demonstrating the procedure.

Applications of the Gram-Schmidt process, defining orthogonal projections, testing for bases, and the significance of orthogonal sets in vector spaces.

Concepts of orthonormal sets and the Gram-Schmidt process for constructing orthonormal bases, including exercises demonstrating the procedure.

Applications of the Gram-Schmidt process, defining orthogonal projections, testing for bases, and the significance of orthogonal sets in vector spaces.

Concepts of orthonormal sets and the Gram-Schmidt process for constructing orthonormal bases, including exercises demonstrating the procedure.

Applications of the Gram-Schmidt process, defining orthogonal projections, testing for bases, and the significance of orthogonal sets in vector spaces.

Concepts of orthonormal sets and the Gram-Schmidt process for constructing orthonormal bases, including exercises demonstrating the procedure.

Applications of the Gram-Schmidt process, defining orthogonal projections, testing for bases, and the significance of orthogonal sets in vector spaces.

Concepts of orthonormal sets and the Gram-Schmidt process for constructing orthonormal bases, including exercises demonstrating the procedure.

Inner product spaces

  • 1.
  • 2.
    11 - 2 Today ◼Inner Product Spaces ◼ Orthonormal Bases:Gram-Schmidt Process ◼ Mathematical Models and Least Square Analysis ◼ Applications
  • 3.
    11 - 3 InnerProduct Spaces (1) (2) (3) (4) and if and only if 〉〈〉〈 uvvu ,, = 〉〈〉〈〉〈 wuvuwvu ,,, +=+ 〉〈〉〈 vuvu ,, cc = 0, 〉〈 vv 0, =〉〈 vv 0=v ◼ Inner product: Let u, v, and w be vectors in a vector space V, and let c be any scalar. An inner product on V is a function that associates a real number <u, v> with each pair of vectors u and v and satisfies the following axioms.
  • 4.
    11 - 4 ◼Note: V Rn spacefor vectorproductinnergeneral, )forproductinnerEuclidean(productdot = = vu vu ◼ Note: A vector space V with an inner product is called an inner product space. ( )•+,,VVector space: Inner product space: ( )•+ ,,,,V
  • 5.
    11 - 5 ◼Ex 1: (The Euclidean inner product for Rn) Show that the dot product in Rn satisfies the four axioms of an inner product. nnvuvuvu +++== 2211, vuvu 〉〈 ),,,(,),,,( 2121 nn vvvuuu  == vu Sol: By Theorem 5.3, this dot product satisfies the required four axioms. Thus it is an inner product on Rn.
  • 6.
    11 - 6 ◼Ex 2: (A different inner product for Rn) Show that the function defines an inner product on R2, where and . 2211 2, vuvu +=〉〈 vu ),(),( 2121 vvuu == vu Sol: 〉〈〉〈 uvvu ,22,)( 22112211 =+=+= uvuvvuvua 〉〈〉〈 〉〈 wuvu wvu ,, )2()2( 22 )(2)(, 22112211 22221111 222111 += +++= +++= +++=+ wuwuvuvu wuvuwuvu wvuwvu ),()( 21 wwb =w
  • 7.
    11 - 7 ◼Note: (An inner product on Rn) 0,, 222111 +++= innn cvucvucvuc 〉〈 vu 〉〈〉〈 vuvu ,)(2)()2(,)( 22112211 cvcuvcuvuvuccc =+=+= 02,)( 2 2 2 1 += vvd 〉〈 vv )0(0020, 21 2 2 2 1 ====+= vvv vvvv〉〈
  • 9.
    11- 9 ◼ Ex3: (A function that is not an inner product) Show that the following function is not an inner product on R3. 332211 2 vuvuvu +−= 〉〈 vu Sol: Let )1,2,1(=v 06)1)(1()2)(2(2)1)(1(,Then −=+−= vv Axiom 4 is not satisfied. Thus this function is not an inner product on R3.
  • 10.
    11- 10 ◼ Thm:(Properties of inner products) Let u, v, and w be vectors in an inner product space V, and let c be any real number. (1) (2) (3) 0,, == 〉〈〉〈 0vv0 〉〈〉〈〉〈 wvwuwvu ,,, +=+ 〉〈〉〈 vuvu ,, cc = ◼ Norm (length) of u: 〉〈 uuu ,|||| = 〉〈 uuu ,|||| 2 = ◼ Note:
  • 11.
    11 - 11 uand v are orthogonal if . ◼ Distance between u and v: −−=−= vuvuvuvu ,||||),(d ◼ Angle between two nonzero vectors u and v:  = 0, |||||||| , cos vu vu 〉〈 ◼ Orthogonal: 0, =〉〈 vu )( vu ⊥
  • 12.
    11- 12 ◼ Notes: (1)If , then v is called a unit vector. (2) 1|||| =v 0 1   v v ⎯⎯⎯ →⎯ gNormalizin v v (the unit vector in the direction of v) not a unit vector
  • 13.
    11 - 13 ◼Ex 6: (Finding inner product) )(inspolynomialbe24)(,21)(Let 2 22 xPxxxqxxp +−=−= nnbababaqp +++= 1100〉,〈 is an inner product ?,)( =〉〈 qpa ?||||)( =qb ?),()( =qpdc Sol: 2)1)(2()2)(0()4)(1(,)( =−+−+=〉〈 qpa 211)2(4,||||)( 222 =+−+== 〉〈 qqqb 22)3(2)3( ,||||),( 323)( 222 2 =−++−= −−=−= −+−=− qpqpqpqpd xxqpc 
  • 15.
    11 - 15 ◼Properties of norm: (1) (2) if and only if (3) ◼ Properties of distance: (1) (2) if and only if (3) 0|||| u 0|||| =u 0u = |||||||||| uu cc = 0),( vud 0),( =vud vu = ),(),( uvvu dd =
  • 16.
    11 - 16 ◼Thm： Let u and v be vectors in an inner product space V. (1) Cauchy-Schwarz inequality: (2) Triangle inequality: (3) Pythagorean theorem : u and v are orthogonal if and only if |||||||||||| vuvu ++ Theorem 5.5 222 |||||||||||| vuvu +=+ Theorem 5.6 |||||||||,| vuvu 〉〈 Theorem 5.4
  • 17.
    11 - 17 ◼Orthogonal projections in inner product spaces: Let u and v be two vectors in an inner product space V, such that . Then the orthogonal projection of u onto v is given by 0v  v vv vu uv   = , , proj ◼ Note: If v is a init vector, then . The formula for the orthogonal projection of u onto v takes the following simpler form. 1||||, 2 == vvv 〉〈 vvuuv = ,proj
  • 18.
    11 - 18 ◼Ex 10: (Finding an orthogonal projection in R3) Use the Euclidean inner product in R3 to find the orthogonal projection of u=(6, 2, 4) onto v=(1, 2, 0). Sol: 10)0)(4()2)(2()1)(6(, =++= vu 5021, 222 =++= vv )0,4,2()0,2,1(proj 5 10 ==   = v vv vu uv
  • 19.
    11 - 19 ◼Thm: (Orthogonal projection and distance) Let u and v be two vectors in an inner product space V, such that . Then0v     vv vu vuuu v , , ,),()proj,( ccdd
  • 20.
    11 - 20 Keywords ◼inner product: 內積 ◼ inner product space: 內積空間 ◼ norm: 範數 ◼ distance: 距離 ◼ angle: 夾角 ◼ orthogonal: 正交 ◼ unit vector: 單位向量 ◼ normalizing: 單範化 ◼ Cauchy – Schwarz inequality: 科西 - 舒瓦茲不等式 ◼ triangle inequality: 三角不等式 ◼ Pythagorean theorem: 畢氏定理 ◼ orthogonal projection: 正交投影
  • 21.
    11 - 21 Today ◼Length and Dot Product in Rn(Continued) ◼ Inner Product Spaces ◼ Orthonormal Bases:Gram-Schmidt Process ◼ Mathematical Models and Least Square Analysis ◼ Applications
  • 22.
    11 - 22 OrthonormalBases: Gram-Schmidt Process ◼ Orthogonal: A set S of vectors in an inner product space V is called an orthogonal set if every pair of vectors in the set is orthogonal. ◼ Orthonormal: An orthogonal set in which each vector is a unit vector is called orthonormal.       = = = ji ji VS ji n 0 1 , ,,, 21 vv vvv   0, ,,, 21 = = ji n VS vv vvv  ji  ◼ Note: If S is a basis, then it is called an orthogonal basis or an orthonormal basis.
  • 23.
    5 - 23 ◼Ex 1: (A nonstandard orthonormal basis for R3) Show that the following set is an orthonormal basis.          −        −          = 3 1 , 3 2 , 3 2 , 3 22 , 6 2 , 6 2 ,0, 2 1 , 2 1 321 S vvv Sol: Show that the three vectors are mutually orthogonal. 0 9 22 9 2 9 2 00 23 2 23 2 00 32 31 6 1 6 1 21 =+−−= =+−= =++−= vv vv vv
  • 24.
    11 - 24 Showthat each vector is of length 1. Thus S is an orthonormal set. 1|||| 1|||| 10|||| 9 1 9 4 9 4 333 9 8 36 2 36 2 222 2 1 2 1 111 =++== =++== =++== vvv vvv vvv
  • 25.
    11 - 25 Thestandard basis is orthonormal. ◼ Ex 2: (An orthonormal basis for ) In , with the inner product )(3 xP 221100, bababaqp ++= },,1{ 2 xxB = )(3 xP Sol: ,001 2 1 xx ++=v ,00 2 2 xx ++=v ,00 2 3 xx ++=v 0)1)(0()0)(1()0)(0(, ,0)1)(0()0)(0()0)(1(, ,0)0)(0()1)(0()0)(1(, 32 31 21 =++= =++= =++= vv vv vv Then
  • 26.
    11 - 26 ()( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 1110000 ,1001100 ,1000011 333 222 111 =++== =++== =++== v,vv v,vv v,vv
  • 27.
    11 - 27 ◼Thm : (Orthogonal sets are linearly independent) If is an orthogonal set of nonzero vectors in an inner product space V, then S is linearly independent.  nS v,,v,v 21= Pf: S is an orthogonal set of nonzero vectors 0and0i.e. = iiji ji v,vv,v iccc ccc iinn nn ==+++ =+++ 0,0, 0Let 2211 2211 vvvvv vvv   = +++++ iii inniiiii c cccc v,v v,vv,vv,vv,v 2211  t.independenlinearlyis00 Siciii = v,v
  • 28.
    11 - 28 ◼Corollary to Thm: If V is an inner product space of dimension n, then any orthogonal set of n nonzero vectors is a basis for V.
  • 29.
    11 - 29 ◼Ex 4: (Using orthogonality to test for a basis) Show that the following set is a basis for .4 R )}1,1,2,1(,)1,2,0,1(,)1,0,0,1(,)2,2,3,2{( 4321 −−−−=S vvvv Sol: : nonzero vectors 02262 02402 02002 41 31 21 =−−+−= =−++−= =−++= vv vv vv 4321 ,,, vvvv 01201 01001 01001 43 42 32 =+−+= =+++−= =+++−= vv vv vv .orthogonalisS 4 forbasisais RS (by Corollary to Theorem 5.10)
  • 30.
    11 - 30 ◼Thm: (Coordinates relative to an orthonormal basis) If is an orthonormal basis for an inner product space V, then the coordinate representation of a vector w with respect to B is },,,{ 21 nB vvv = },,,{ 21 nB vvv  = is orthonormal ji ji ji  =    = 0 1 , vv Vw nnkkk vvvw +++= 2211 (unique representation) Pf: is a basis for V},,,{ 21 nB vvv = nn vvwvvwvvww +++= ,,, 2211 
  • 31.
    11 - 31 ik kkk kkk i inniiii inni = ++++= +++= 〉,〈〉,〈〉,〈 〉,)(〈〉,〈 11 2211 vvvvvv vvvvvw   nnvvwvvwvvww +++= ,,, 2211  ◼ Note: If is an orthonormal basis for V and ,},,,{ 21 nB vvv = Vw Then the corresponding coordinate matrix of w relative to B is                  = n B vw vw vw w , , , 2 1 
  • 32.
    11 - 32 ◼Ex 5: (Representing vectors relative to an orthonormal basis) Find the coordinates of w = (5, -5, 2) relative to the following orthonormal basis for . )}1,0,0(,)0,,(,)0,,{( 5 3 5 4 5 4 5 3 −=B 3 R Sol: 2)1,0,0()2,5,5(, 7)0,,()2,5,5(, 1)0,,()2,5,5(, 33 5 3 5 4 22 5 4 5 3 11 =−== −=−−== −=−== vwvw vwvw vwvw           − − = 2 7 1 ][ Bw
  • 33.
    11 - 33 ◼Gram-Schmidt orthonormalization process: is a basis for an inner product space V},,,{ 21 nB uuu = 11Let uv = })({1 1vw span= }),({2 21 vvw span= },,,{' 21 nB vvv = },,,{'' 2 2 n n B v v v v v v 1 1 = is an orthogonal basis. is an orthonormal basis.  − = −=−= − 1 1 〉〈 〉〈 proj 1 n i i ii in nnnn n v v,v v,v uuuv W  2 22 23 1 11 13 3333 〉〈 〉〈 〉〈 〉〈 proj 2 v v,v v,u v v,v v,u uuuv W −−=−= 1 11 12 2222 〉〈 〉〈 proj 1 v v,v v,u uuuv W −=−=
  • 34.
    11 - 34 Sol:)0,1,1(11 == uv )2,0,0()0, 2 1 , 2 1 ( 2/1 2/1 )0,1,1( 2 1 )2,1,0( 2 22 23 1 11 13 33 =−−−=   −   −= v vv vu v vv vu uv ◼ Ex 7: (Applying the Gram-Schmidt orthonormalization process) Apply the Gram-Schmidt process to the following basis. )}2,1,0(,)0,2,1(,)0,1,1{( 321 =B uuu )0, 2 1 , 2 1 ()0,1,1( 2 3 )0,2,1(1 11 12 22 −=−=   −= v vv vu uv
  • 35.
    11 - 35 }2)0,(0,0),, 2 1 , 2 1 (0),1,(1,{},,{'321 − == vvvB Orthogonal basis }1)0,(0,0),, 2 1 , 2 1 (0),, 2 1 , 2 1 ({},,{'' 3 3 2 2 − == v v v v v v 1 1 B Orthonormal basis
  • 36.
    11 - 36 ◼Ex 10: (Alternative form of Gram-Schmidt orthonormalization process) Find an orthonormal basis for the solution space of the homogeneous system of linear equations. 0622 07 4321 421 =+++ =++ xxxx xxx Sol:       − − ⎯⎯ →⎯      08210 01201 06212 07011 .. EJG             − +            − =             − +− =              1 0 8 1 0 1 2 2 82 2 4 3 2 1 ts t s ts ts x x x x
  • 37.
    11 - 37 Thusone basis for the solution space is )}1,0,8,1(,)0,1,2,2{(},{ 21 −−== vvB ( ) ( ) ( ) ( )1,2,4,3 01,2,,2 9 18 10,8,1, , , 01,2,,2 1 11 12 22 11 −−= − − −−=   −= −== v vv vu uv uv ( )( ) 1,2,4,30,1,2,2' −−−= B (orthogonal basis)             −−       − = 30 1 , 30 2 , 30 4 , 30 3 ,0, 3 1 , 3 2 , 3 2 ''B (orthonormal basis)
  • 38.
    11 - 38 Keywords: ◼orthogonal set: 正交集合 ◼ orthonormal set: 單範正交集合 ◼ orthogonal basis: 正交基底 ◼ orthonormal basis: 單範正交基底 ◼ linear independent: 線性獨立 ◼ Gram-Schmidt Process: Gram-Schmidt過程
  • 39.