程序代写代做代考 algorithm kernel 10.1-10.2

10.1-10.2
Week 11
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 11

Outline
1 10.1-10.2
10.1-10.2
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V .
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V . (4)⟨rv,w⟩=r⟨v,w⟩forallv,w∈V andr∈R.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V . (4)⟨rv,w⟩=r⟨v,w⟩forallv,w∈V andr∈R. (5) ⟨v,v⟩ > 0 for all non-zero v ∈ V.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V . (4)⟨rv,w⟩=r⟨v,w⟩forallv,w∈V andr∈R. (5) ⟨v,v⟩ > 0 for all non-zero v ∈ V.
A real vector space V with an inner product ⟨, ⟩ will be called an inner product space.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V . (4)⟨rv,w⟩=r⟨v,w⟩forallv,w∈V andr∈R. (5) ⟨v,v⟩ > 0 for all non-zero v ∈ V.
A real vector space V with an inner product ⟨, ⟩ will be called an inner product space.Note that every subspace of an inner product space is again an inner product space using the same inner product.
Week 11
Ali Mousavidehshikh

10.1-10.2
Definition: An inner product on a real vector space V is a function that assigns a real number ⟨v,w⟩ to every pair of vectors v , w ∈ V satisfying the following axioms:
(1) ⟨v,w⟩ ∈ R for all v,w ∈ V.
(2) ⟨v,w⟩ = ⟨w,v⟩ for all v,w ∈ V.
(3) ⟨v + w , u⟩ = ⟨v , u⟩ for all u, v , w ∈ V . (4)⟨rv,w⟩=r⟨v,w⟩forallv,w∈V andr∈R. (5) ⟨v,v⟩ > 0 for all non-zero v ∈ V.
A real vector space V with an inner product ⟨, ⟩ will be called an inner product space.Note that every subspace of an inner product space is again an inner product space using the same inner product.
Example: the usual dot product is an inner product on Rn.
Ali Mousavidehshikh
Week 11

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R.
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R. (2) ⟨v,0⟩ = 0 = ⟨0,v⟩ for any v ∈ V.
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R. (2) ⟨v,0⟩ = 0 = ⟨0,v⟩ for any v ∈ V.
(3) ⟨v,v⟩ = 0 if and only if v = 0.
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R.
(2) ⟨v,0⟩ = 0 = ⟨0,v⟩ for any v ∈ V.
(3) ⟨v,v⟩ = 0 if and only if v = 0.
(4) 􏰀ni=1⟨riui,w⟩ = 􏰀ni=1 ri⟨ui,w⟩ (similar result in the other coordinate).
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R.
(2) ⟨v,0⟩ = 0 = ⟨0,v⟩ for any v ∈ V.
(3) ⟨v,v⟩ = 0 if and only if v = 0.
(4) 􏰀ni=1⟨riui,w⟩ = 􏰀ni=1 ri⟨ui,w⟩ (similar result in the other coordinate).
If A is any n × n positive definite matrix, then ⟨x , y ⟩ = x T Ay for all x,y ∈ Rn (columns) defines an inner product on Rn, and every inner product on Rn arises this way.
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: (1)⟨u,v+w⟩=⟨u,v⟩+⟨u,w⟩and
⟨v,rw⟩ = r⟨v,w⟩ = ⟨rv,w⟩ for all u,v,w ∈ V and r ∈ R.
(2) ⟨v,0⟩ = 0 = ⟨0,v⟩ for any v ∈ V.
(3) ⟨v,v⟩ = 0 if and only if v = 0.
(4) 􏰀ni=1⟨riui,w⟩ = 􏰀ni=1 ri⟨ui,w⟩ (similar result in the other coordinate).
If A is any n × n positive definite matrix, then ⟨x , y ⟩ = x T Ay for all x,y ∈ Rn (columns) defines an inner product on Rn, and every inner product on Rn arises this way.
Let the inner product ⟨, ⟩ be defined on R2 by ⟨(x,y)T,(u,v)T⟩=2xu−xv−yu−yv. Then
TTT 􏰑2−1􏰒
⟨(x,y) ,(u,v) ⟩=x Ay,whereA= −1 1 . Notice
that A is a positive definite matrix.
Ali Mousavidehshikh
Week 11

10.1-10.2
Norm: ∥ v ∥= 􏰟⟨v , v ⟩, Distance: d (v , w ) =∥ v − w ∥. A vector is called a unit vector if and only if ∥ v ∥= 1. It follows
that for any non-zero v ∈ V , v is a unit vector. ∥v∥
Week 11
Ali Mousavidehshikh

10.1-10.2
Norm: ∥ v ∥= 􏰟⟨v , v ⟩, Distance: d (v , w ) =∥ v − w ∥. A vector is called a unit vector if and only if ∥ v ∥= 1. It follows
that for any non-zero v ∈ V , v is a unit vector. ∥v∥
Example: Show that ⟨u + v , u − v ⟩ =∥ u ∥2 − ∥ v ∥2 .
Week 11
Ali Mousavidehshikh

10.1-10.2
Norm: ∥ v ∥= 􏰟⟨v , v ⟩, Distance: d (v , w ) =∥ v − w ∥. A vector is called a unit vector if and only if ∥ v ∥= 1. It follows
that for any non-zero v ∈ V , v is a unit vector. ∥v∥
Example: Show that ⟨u + v , u − v ⟩ =∥ u ∥2 − ∥ v ∥2 .
Cauchy-Schwartz Inequality: ⟨v,w⟩2 ≤∥ v ∥2∥ w ∥2. Moreover, equality holds if and only if one of v and w is a scalar multiple of the other.
Ali Mousavidehshikh
Week 11

10.1-10.2
Norm: ∥ v ∥= 􏰟⟨v , v ⟩, Distance: d (v , w ) =∥ v − w ∥. A vector is called a unit vector if and only if ∥ v ∥= 1. It follows
that for any non-zero v ∈ V , v is a unit vector. ∥v∥
Example: Show that ⟨u + v , u − v ⟩ =∥ u ∥2 − ∥ v ∥2 .
Cauchy-Schwartz Inequality: ⟨v,w⟩2 ≤∥ v ∥2∥ w ∥2. Moreover, equality holds if and only if one of v and w is a scalar multiple of the other.
Proof . Let ∥ v ∥= a and ∥ w ∥= b. Then
∥bv−aw∥2 = 2ab(ab−⟨v,w⟩)and (1) ∥bv+aw∥2 = 2ab(ab+⟨v,w⟩). (2)
It follows that ab − ⟨v , w ⟩ ≥ 0 and ab + ⟨v , w ⟩ ≥ 0, and hence −ab ≤ ⟨v,w⟩ ≤ ab. That is, |⟨v,w⟩| ≤ ab, as desired. The second part is an exercise.
Ali Mousavidehshikh
Week 11

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
aaa
Week 11
Ali Mousavidehshikh

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
aaa
Week 11
Ali Mousavidehshikh

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
(2) ∥ v ∥= 0 if and only if v = 0.
aaa
Week 11
Ali Mousavidehshikh

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
(2) ∥ v ∥= 0 if and only if v = 0. (3)∥rv∥=|r| ∥v∥.
aaa
Week 11
Ali Mousavidehshikh

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
(2) ∥ v ∥= 0 if and only if v = 0. (3)∥rv∥=|r| ∥v∥.
(4) ∥ v + w ∥≤∥ v ∥ + ∥ w ∥ (triangle inequality).
aaa
Week 11
Ali Mousavidehshikh

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
(2) ∥ v ∥= 0 if and only if v = 0. (3)∥rv∥=|r| ∥v∥.
(4) ∥ v + w ∥≤∥ v ∥ + ∥ w ∥ (triangle inequality).
Let ⟨v1,…,vn} be a spanning set for V. If v ∈ V and ⟨v,vi⟩ = 0 for each i = 1,…,n, then v = 0.
aaa
Ali Mousavidehshikh
Week 11

10.1-10.2
Example: If f and g are continuous functions on the interval [a, b], then (􏰦 b f (x)g(x)dx)2 ≤ (􏰦 b f (x)dx)2(􏰦 b g(x)dx)2.
Theorem: For any v,w ∈ V and r ∈ R, (1) ∥ v ∥≥ 0 for every v ∈ V .
(2) ∥ v ∥= 0 if and only if v = 0. (3)∥rv∥=|r| ∥v∥.
(4) ∥ v + w ∥≤∥ v ∥ + ∥ w ∥ (triangle inequality).
Let ⟨v1,…,vn} be a spanning set for V. If v ∈ V and ⟨v,vi⟩ = 0 for each i = 1,…,n, then v = 0.
Solution. Let v = 􏰀ni=1 rivi. Then
∥ v ∥2= ⟨v,v⟩ = ⟨v,􏰀ni=1 rivi⟩ = 0.
aaa
Ali Mousavidehshikh
Week 11

Theorem: For any u,v,w ∈ V, (1) d(v,w) ≥ 0.
10.1-10.2
Week 11
Ali Mousavidehshikh

Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w.
10.1-10.2
Week 11
Ali Mousavidehshikh

Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w. (3) d(v,w) = d(w,v).
10.1-10.2
Week 11
Ali Mousavidehshikh

Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w. (3) d(v,w) = d(w,v).
(4) d(v, w) ≤ d(v, u) + d(u, w).
10.1-10.2
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w. (3) d(v,w) = d(w,v).
(4) d(v, w) ≤ d(v, u) + d(u, w).
Determine if the following is an inner product on R2: ⟨(x,y),(u,v)⟩ = xyuv.
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w. (3) d(v,w) = d(w,v).
(4) d(v, w) ≤ d(v, u) + d(u, w).
Determine if the following is an inner product on R2: ⟨(x,y),(u,v)⟩ = xyuv.
Define⟨u,v⟩=u·v onR4. Findd((1,2,3,4),(0,−1,2,0)).
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: For any u,v,w ∈ V,
(1) d(v,w) ≥ 0.
(2) d(v,w) = 0 if and only if v = w. (3) d(v,w) = d(w,v).
(4) d(v, w) ≤ d(v, u) + d(u, w).
Determine if the following is an inner product on R2: ⟨(x,y),(u,v)⟩ = xyuv.
Define⟨u,v⟩=u·v onR4. Findd((1,2,3,4),(0,−1,2,0)).
If T : V → V is an isomorphism of the inner product space V with inner product ⟨, ⟩, define a new product
⟨v,w⟩1 = ⟨T(v),T(w)⟩. Show that ⟨,⟩1 is an inner product on V.
Ali Mousavidehshikh
Week 11

10.1-10.2
A set {f1,f2,…,fn} of vectors is an orthogonal set if fi ̸= 0 for each i = 1,2,…,n, and ⟨fi,fj⟩ = 0 for all i ̸= j. The set is called orthonormal if in addition ∥ fi ∥= 1 for each
i = 1,2,…,n.
Week 11
Ali Mousavidehshikh

10.1-10.2
A set {f1,f2,…,fn} of vectors is an orthogonal set if fi ̸= 0 for each i = 1,2,…,n, and ⟨fi,fj⟩ = 0 for all i ̸= j. The set is called orthonormal if in addition ∥ fi ∥= 1 for each
i = 1,2,…,n.
Let C[a,b] be the set of continuous functions from [a,b] to
R. Then C[a,b] is a subspace of F[a,b]. The product
⟨f,g⟩ = 􏰦b f(x)g(x)dx defines an inner product on C[a,b] a
(exercise).
Week 11
Ali Mousavidehshikh

10.1-10.2
A set {f1,f2,…,fn} of vectors is an orthogonal set if fi ̸= 0 for each i = 1,2,…,n, and ⟨fi,fj⟩ = 0 for all i ̸= j. The set is called orthonormal if in addition ∥ fi ∥= 1 for each
i = 1,2,…,n.
Let C[a,b] be the set of continuous functions from [a,b] to
R. Then C[a,b] is a subspace of F[a,b]. The product
⟨f,g⟩ = 􏰦b f(x)g(x)dx defines an inner product on C[a,b] a
(exercise).
The set {sinx,cosx} is an orthogonal set in C[−π,π] because
􏰦π sinxcosxdx=0. −π
Ali Mousavidehshikh
Week 11

10.1-10.2
Theorem: If {f1, f2, . . . , fn} is an orthogonal set of vectors, then ∥ 􏰀ni=1 fi ∥2= 􏰀ni=1 ∥ fi ∥2. Moreover,
{r1f1, r2f2, . . . , rnfn} is also an orthogonal set for any ri ̸= 0 in
􏰖 f1 fn 􏰤
R. Furthermore, ∥ f1 ∥,…, ∥ fn ∥ is an orthonormal set
(this is called normalizing the set {f1 , . . . , fn }).
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: If {f1, f2, . . . , fn} is an orthogonal set of vectors, then ∥ 􏰀ni=1 fi ∥2= 􏰀ni=1 ∥ fi ∥2. Moreover,
{r1f1, r2f2, . . . , rnfn} is also an orthogonal set for any ri ̸= 0 in
􏰖 f1 fn 􏰤
R. Furthermore, ∥ f1 ∥,…, ∥ fn ∥ is an orthonormal set
(this is called normalizing the set {f1 , . . . , fn }).    200
Example Show that −1 , 1 , −1 is an 012
orthogonal basis for R2 with inner product [v , w ] = v T Aw 1 1 0
where A=1 2 0. 001
Week 11
Ali Mousavidehshikh

10.1-10.2
Theorem: If {f1, f2, . . . , fn} is an orthogonal set of vectors, then ∥ 􏰀ni=1 fi ∥2= 􏰀ni=1 ∥ fi ∥2. Moreover,
{r1f1, r2f2, . . . , rnfn} is also an orthogonal set for any ri ̸= 0 in
􏰖 f1 fn 􏰤
R. Furthermore, ∥ f1 ∥,…, ∥ fn ∥ is an orthonormal set
(this is called normalizing the set {f1 , . . . , fn }).    200
Example Show that −1 , 1 , −1 is an 012
orthogonal basis for R2 with inner product [v , w ] = v T Aw 1 1 0
where A=1 2 0. 001
Expansion Theorem: Let {f1, . . . , fn} be an orthogonal basis
ofaninnerproductspaceV.Ifv∈V,thenv=􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Ali Mousavidehshikh
Week 11

10.1-10.2
Orthogonal lemma: Let {f1, . . . , fm} be an orthogonal set of vectors in an inner product space V , and let v ∈ V such that v ∈/ span{f1,…,fm}, then {f1,…,fm,fm+1} is an orthogonal
set,wherefm+1=v−􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Week 11
Ali Mousavidehshikh

10.1-10.2
Orthogonal lemma: Let {f1, . . . , fm} be an orthogonal set of vectors in an inner product space V , and let v ∈ V such that v ∈/ span{f1,…,fm}, then {f1,…,fm,fm+1} is an orthogonal
set,wherefm+1=v−􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Gram-Schmidt Orthogonalization Algorithm: Let V be an inner product space and let {v1,…,vn} be any basis of V. Define f1, . . . , fn in V successively as follows: f1 = v1 and
k−1 ⟨vk,fi⟩
fk =vk −􏰁 fi foreachk=2,3,…,n. Then:
i=1 ∥fi ∥2
Week 11
Ali Mousavidehshikh

10.1-10.2
Orthogonal lemma: Let {f1, . . . , fm} be an orthogonal set of vectors in an inner product space V , and let v ∈ V such that v ∈/ span{f1,…,fm}, then {f1,…,fm,fm+1} is an orthogonal
set,wherefm+1=v−􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Gram-Schmidt Orthogonalization Algorithm: Let V be an inner product space and let {v1,…,vn} be any basis of V. Define f1, . . . , fn in V successively as follows: f1 = v1 and
k−1 ⟨vk,fi⟩
fk =vk −􏰁 fi foreachk=2,3,…,n. Then:
i=1 ∥fi ∥2
(1) {f1,…,fn} is an orthogonal basis of V,
Week 11
Ali Mousavidehshikh

10.1-10.2
Orthogonal lemma: Let {f1, . . . , fm} be an orthogonal set of vectors in an inner product space V , and let v ∈ V such that v ∈/ span{f1,…,fm}, then {f1,…,fm,fm+1} is an orthogonal
set,wherefm+1=v−􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Gram-Schmidt Orthogonalization Algorithm: Let V be an inner product space and let {v1,…,vn} be any basis of V. Define f1, . . . , fn in V successively as follows: f1 = v1 and
k−1 ⟨vk,fi⟩
fk =vk −􏰁 fi foreachk=2,3,…,n. Then:
i=1 ∥fi ∥2
(1) {f1,…,fn} is an orthogonal basis of V,
(2) span{f1,…,fk} = span{v1,…,vk} for each k = 1,2,…,n.
Week 11
Ali Mousavidehshikh

10.1-10.2
Orthogonal lemma: Let {f1, . . . , fm} be an orthogonal set of vectors in an inner product space V , and let v ∈ V such that v ∈/ span{f1,…,fm}, then {f1,…,fm,fm+1} is an orthogonal
set,wherefm+1=v−􏰁n ⟨v,fi⟩fi. i=1 ∥fi ∥2
Gram-Schmidt Orthogonalization Algorithm: Let V be an inner product space and let {v1,…,vn} be any basis of V. Define f1, . . . , fn in V successively as follows: f1 = v1 and
k−1 ⟨vk,fi⟩
fk =vk −􏰁 fi foreachk=2,3,…,n. Then:
(1) {f1,…,fn} is an orthogonal basis of V,
(2) span{f1,…,fk} = span{v1,…,vk} for each
k = 1,2,…,n.
Example. Consider V = P3 with the inner product
⟨p, q⟩ = 􏰦 1 p(x)q(x)dx. Apply the Gram-Schmidt method to −1
the standard basis {1,x,x2,x3} to get an orthogonal basis for P3.
i=1 ∥fi ∥2
Ali Mousavidehshikh
Week 11

10.1-10.2
If V is an inner product space of dimension n, then
CE : V → Rn is an isomorphism (where E = {f1,…,fn} is any orthonormal basis for V ). Moreover,
⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space.
Week 11
Ali Mousavidehshikh

10.1-10.2
If V is an inner product space of dimension n, then
CE : V → Rn is an isomorphism (where E = {f1,…,fn} is any orthonormal basis for V ). Moreover,
⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space.
Orthogonal complement:
U⊥ = {v ∈ V : ⟨v,u⟩ = 0 for all u ∈ U}.
Week 11
Ali Mousavidehshikh

10.1-10.2
If V is an inner product space of dimension n, then
CE : V → Rn is an isomorphism (where E = {f1,…,fn} is any orthonormal basis for V ). Moreover,
⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space.
Orthogonal complement:
U⊥ = {v ∈ V : ⟨v,u⟩ = 0 for all u ∈ U}.
Theorem: (1)U⊥ isasubspaceofV andV =U⊕U⊥.
Week 11
Ali Mousavidehshikh

10.1-10.2
If V is an inner product space of dimension n, then
CE : V → Rn is an isomorphism (where E = {f1,…,fn} is any orthonormal basis for V ). Moreover,
⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space.
Orthogonal complement:
U⊥ = {v ∈ V : ⟨v,u⟩ = 0 for all u ∈ U}.
Theorem: (1)U⊥ isasubspaceofV andV =U⊕U⊥. (2)IfdimV =n,thendimU+dimU⊥ =n.
Week 11
Ali Mousavidehshikh

10.1-10.2
If V is an inner product space of dimension n, then
CE : V → Rn is an isomorphism (where E = {f1,…,fn} is any orthonormal basis for V ). Moreover,
⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space.
Orthogonal complement:
U⊥ = {v ∈ V : ⟨v,u⟩ = 0 for all u ∈ U}.
Theorem: (1)U⊥ isasubspaceofV andV =U⊕U⊥. (2)IfdimV =n,thendimU+dimU⊥ =n.
(3)IfdimV <∞,thenU⊥⊥ =U. Week 11 Ali Mousavidehshikh 10.1-10.2 If V is an inner product space of dimension n, then CE : V → Rn is an isomorphism (where E = {f1,...,fn} is any orthonormal basis for V ). Moreover, ⟨v,w⟩ = ⟨􏰀ni=1 aifi,􏰀ni=1 bifi⟩ = 􏰀i,j aibi⟨fi,fj⟩ = 􏰀ni=1aibi =CE(v)·CE(w). Thatis,CE preservesinner products. Hence, V is isomorphic to Rn as an inner product space. Orthogonal complement: U⊥ = {v ∈ V : ⟨v,u⟩ = 0 for all u ∈ U}. Theorem: (1)U⊥ isasubspaceofV andV =U⊕U⊥. (2)IfdimV =n,thendimU+dimU⊥ =n. (3)IfdimV <∞,thenU⊥⊥ =U. The projection of U with kernel U⊥ is called the orthogonal projection on U and is denoted by projU : V → V defined via the following process: since V = U ⊕ U⊥, given x ∈ V , x = u + w for some u ∈ U and w ∈ U⊥. We define projU(x) = u. Ali Mousavidehshikh Week 11 10.1-10.2 Projection Theorem: Let U be a finite dimensional subspace of an inner product space V . (1) projU : V → V is a linear operator with image U and kernel U⊥. Week 11 Ali Mousavidehshikh 10.1-10.2 Projection Theorem: Let U be a finite dimensional subspace of an inner product space V . (1) projU : V → V is a linear operator with image U and kernel U⊥. (2)projU(v)∈Uandv−projU(v)∈U⊥ forallv∈V. Week 11 Ali Mousavidehshikh 10.1-10.2 Projection Theorem: Let U be a finite dimensional subspace of an inner product space V . (1) projU : V → V is a linear operator with image U and kernel U⊥. (2)projU(v)∈Uandv−projU(v)∈U⊥ forallv∈V. (3) If {f1, . . . , fm} is any orthogonal basis of U, then for any v∈Vwehaveproj (v)=􏰀m ⟨v,fi⟩f. U i=1 ∥ fi ∥2 i Week 11 Ali Mousavidehshikh 10.1-10.2 Projection Theorem: Let U be a finite dimensional subspace of an inner product space V . (1) projU : V → V is a linear operator with image U and kernel U⊥. (2)projU(v)∈Uandv−projU(v)∈U⊥ forallv∈V. (3) If {f1, . . . , fm} is any orthogonal basis of U, then for any v∈Vwehaveproj (v)=􏰀m ⟨v,fi⟩f. U i=1 ∥ fi ∥2 i Approximation Theorem: Let U be a finite dimensional subspace of an inner product space V . Given v ∈ V , projU (v ) is the vector in U closest to v. That is, ∥ v − projU (v ) ∥<∥ v − u ∥ for all u ∈ U , u ̸= projU (v ). Ali Mousavidehshikh Week 11 10.1-10.2 Consider the space C[−1,1] of real-valued continuous functions on the interval [−1, 1] with inner product ⟨f,g⟩=􏰦1 f(x)g(x)dx. Findthepolynomialp(x)ofdegree −1 at most 2 that best approximates the absolute values function f (x) = |x|. Ali Mousavidehshikh Week 11 10.1-10.2 Consider the space C[−1,1] of real-valued continuous functions on the interval [−1, 1] with inner product ⟨f,g⟩=􏰦1 f(x)g(x)dx. Findthepolynomialp(x)ofdegree −1 at most 2 that best approximates the absolute values function f (x) = |x|. Solution. Let U = P2. Ali Mousavidehshikh Week 11 10.1-10.2 Consider the space C[−1,1] of real-valued continuous functions on the interval [−1, 1] with inner product ⟨f,g⟩=􏰦1 f(x)g(x)dx. Findthepolynomialp(x)ofdegree −1 at most 2 that best approximates the absolute values function f (x) = |x|. Solution. Let U = P2. We can apply the Gram-Schmidt method to the basis {1, x, x2} to get the orthogonal basis {f1 =1,f2 =x,f3 =3x2−1}ofP2. Ali Mousavidehshikh Week 11 10.1-10.2 Consider the space C[−1,1] of real-valued continuous functions on the interval [−1, 1] with inner product ⟨f,g⟩=􏰦1 f(x)g(x)dx. Findthepolynomialp(x)ofdegree −1 at most 2 that best approximates the absolute values function f (x) = |x|. Solution. Let U = P2. We can apply the Gram-Schmidt method to the basis {1, x, x2} to get the orthogonal basis {f1 =1,f2 =x,f3 =3x2−1}ofP2.Hence, proj (f)=􏰀3 ⟨f,fi⟩f =1f +0f + 5f = 3(5x2+1). P2 i=1 ∥ fi ∥2 i 2 1 2 16 3 16 Ali Mousavidehshikh Week 11