Operators on Inner Product Spaces

大學線性代數再探

大學數學

大 學 線性代數 .

linear operator . 線性代數 , 性 .

linear transformation 性 , 再 . 代數

field 性 over field polynomial ring 代數 (

).

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代. , .

, . , 性

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v

Chapter 5

Operators on Inner Product Spaces

inner product spaces linear operators 性 . inner

product spaces vector spaces , 探

linear operators. inner product spaces, vector

spaces vector space over C R.

5.1. Inner Product Spaces

, inner product space 性 .

Real inner product space.

Definition 5.1.1. V vector space overR. 數 ⟨, ⟩ : V ×V → R 性

, V inner product.

(1) ⟨v,w⟩ = ⟨w,v⟩, ∀v,w ∈ V.

(2) ⟨rv + sw,u⟩ = r⟨v,u⟩ + s⟨w,u⟩, ∀u,v,w ∈ V and r,s ∈ R.

(3) ⟨v,v⟩ ≥ 0, ∀v ∈ V. ⟨v,v⟩ = 0 v = O_V. V real inner product space.

complex , , z∈ C, z z conjugate ( 數).

Definition 5.1.2. V vector space overC. 數 ⟨, ⟩ : V ×V → C 性

, V inner product.

(1) ⟨v,w⟩ = ⟨w,v⟩, ∀v,w ∈ V.

(2) ⟨rv + sw,u⟩ = r⟨v,u⟩ + s⟨w,u⟩, ∀u,v,w ∈ V and r,s ∈ C.

(3) ⟨v,v⟩ ≥ 0, ∀v ∈ V. ⟨v,v⟩ = 0 v = O_V.

105

V complex inner product space.

, vector space inner product. V inner product

space, inner product.

Example 5.1.3. Rⁿ

⟨(x1, . . . , xn), (y1, . . . , yn)⟩ = x1y1+··· + xnyn,

Rⁿ standard inner product. inner product , Rⁿ n-dimensional Euclidean space.

Cⁿ

⟨(x1, . . . , xn), (y1, . . . , yn)⟩ = x1y1+··· + xnyn,

Cⁿ standard inner product. inner product , Cⁿ n-dimensional unitary space.

Question 5.1. V over C inner product space. V vector space over R, V over R inner product space?

real inner product space , (1) 性, (2)

u, v, w∈ V r, s∈ R

⟨u,rv + sw⟩ = r⟨u,v⟩ + s⟨u,w⟩.

v, v^′, w, w^′∈ V r, r^′, s, s^′∈ R

⟨rv + r^′v^′, sw + s^′w^′⟩ = r⟨v,sw + s^′w^′⟩ + r^′⟨v^′, sw + s^′w^′⟩

= rs⟨v,w⟩ + rs^′⟨v,w^′⟩ + r^′s⟨v^′, w⟩ + r^′s^′⟨v^′, w^′⟩ (5.1)

complex , (1), (2) u, v, w∈ V r, s∈ C

⟨u,rv + sw⟩ = ⟨rv + sw,u⟩ = r⟨v,u⟩ + s⟨w,u⟩ = r⟨u,v⟩ + s⟨u,w⟩.

v, v^′, w, w^′∈ V r, r^′, s, s^′∈ C

⟨rv + r^′v^′, sw + s^′w^′⟩ = r⟨v,sw + s^′w^′⟩ + r^′⟨v^′, sw + s^′w^′⟩

= rs⟨v,w⟩ + rs^′⟨v,w^′⟩ + r^′s⟨v^′, w⟩ + r^′s^′⟨v^′, w^′⟩ (5.2) inner product ,⟨v,v⟩ = 0 v = O_V, 性 non-degenerate.

性 .

Lemma 5.1.4. V inner product space v∈ V ⟨v,w⟩ = 0, ∀w ∈ V, v = O_V.

Proof. w = v, ⟨v,v⟩ = 0. inner product v = O_V. 

5.1. Inner Product Spaces 107

Lemma 5.1.4 V O_V . v, u∈ V,

⟨v,w⟩ = ⟨u,w⟩, ∀w ∈ V,

⟨v − u,w⟩ = ⟨v,w⟩ − ⟨u,w⟩ = 0

v = u. 性 .

Corollary 5.1.5. V inner product space. v, u∈ V ⟨v,w⟩ = ⟨u,w⟩, ∀w ∈ V, v = u.

Lemma 5.1.4 linear operator, 性 .

real complex .

Proposition 5.1.6. V inner product space T : V → V linear operator.

(1) V real inner product space, ⟨T(v),w⟩ = 0, ∀v,w ∈ V T zero mapping.

(2) V complex inner product space, ⟨T(v),v⟩ = 0, ∀v ∈ V T zero mapping.

Proof. v∈ V, ⟨T(v),w⟩ = 0, ∀w ∈ V, Lemma 5.1.4 T (v) = OV.

v∈ V , T = O.

complex , (5.2) v, w∈ V r∈ C

0 = ⟨T(rv + w),rv + w⟩

= ⟨rT(v) + T(w),rv + w⟩

= rr⟨T(v),v⟩ + r⟨T(v),w⟩ + r⟨T(w),v⟩ + ⟨T(w),w⟩

= r⟨T(v),w⟩ + r⟨T(w),v⟩

代 r = 1 r =√

−1, ⟨T(v),w⟩ + ⟨T(w),v⟩ = 0 ⟨T(v),w⟩ − ⟨T(w),v⟩ = 0.

⟨T(v),w⟩, ∀v,w ∈ V. 前 T = O. 

V inner product space, over R over C,

Cauchy-Schwarz inequality.

Lemma 5.1.7. V inner product space over F, F =R C.

v∈ V ∥v∥ =√

⟨v,v⟩, v, w∈ V,

|⟨v,w⟩| ≤ ∥v∥∥w∥.

|⟨v,w⟩| = ∥v∥∥w∥ v, w O_V r∈ F v = rw.

Proof. v, w O_V, . v, w O_V.

r∈ F, F R (5.1)

0≤ ⟨v − rw,v − rw⟩ = ⟨v,v⟩ − 2r⟨v,w⟩ + r²⟨w,w⟩.

r =⟨v,w⟩/⟨w,w⟩,

⟨v,w⟩²

⟨w,w⟩ ≤ ⟨v,v⟩.

F C (5.2)

0≤ ⟨v − rw,v − rw⟩ = ⟨v,v⟩ − r⟨w,v⟩ − r⟨v,w⟩ + rr⟨w,w⟩.

r =⟨v,w⟩/⟨w,w⟩ ( r =⟨v,w⟩/⟨w,w⟩, ⟨w,w⟩ ∈ R),

⟨v,w⟩⟨v,w⟩

⟨w,w⟩ ≤ ⟨v,v⟩.

F =C ⟨v,w⟩⟨v,w⟩ = |⟨v,w⟩|² inequality.

r∈ F ⟨v − rw,v − rw⟩ = 0 v = rw. 

inner product , norm. 性

:

Proposition 5.1.8. V inner product space over F (F =R C).

v∈ V ∥v∥ =√

⟨v,v⟩, 性 :

(1) ∥v∥ ≥ 0 ∥v∥ = 0 v = O_V.

(2) r∈ F v∈ V, ∥rv∥ = |r|∥v∥.

(3) v, w∈ V, ∥v + w∥ ≤ ∥v∥ + ∥w∥.

Proof. (1) inner product 性 (3) , (2) (5.1), (5.2) , (3). ⟨v + w,v + w⟩ = ⟨v,v⟩ + 2⟨v,w⟩ + ⟨w,w⟩, Lemma 5.1.7

∥v + w∥²=⟨v + w,v + w⟩ ≤ ∥v∥²+ 2∥v∥∥w∥ + ∥w∥²= (∥v∥ + ∥w∥)²,

∥v + w∥ ≤ ∥v∥ + ∥w∥. 

Proposition 5.1.8 (3) 性 (triangle inequality). vector

space V , 數 ∥ ∥ : V → R Proposition 5.1.8 性 normed

linear space, 數 ∥ ∥ norm. inner product space

Proposition 5.1.8 norm normed linear space. norm

v, w∈ V, v, w (distance) d(v, w) =∥v − w∥. distance vector space metric space. inner product space metric space.

metric space , sequence .

, .

Question 5.2. V inner product space. Proposition 5.1.8 norm parallelogram law, v, w∈ V

∥v + w∥²+∥v − w∥²= 2∥v∥²+ 2∥w∥².

5.1. Inner Product Spaces 109

Inner product space metric space, 性 (or-

thogonal) . .

Definition 5.1.9. V inner product space. v, w∈ V ⟨v,w⟩ = 0, v, w orthogonal, v⊥ w .

{v1, . . . , v_n} V basis i̸= j v_i⊥ vj, {v1, . . . , v_n}

V orthogonal basis. orthogonal basis v_i ∥vi∥ = 1, V

orthonormal basis.

⟨v,w⟩ = 0, ⟨w,v⟩ = 0, v⊥ w w⊥ v.

Question 5.3. ⊥ equivalent relation? equivalent relation ? {w1, . . . , wn} V orthogonal basis, v_i= _∥w¹

i∥w_i, {v1, . . . , vn} V orthonormal basis.

orthogonal basis ( orthonormal basis) {v1, . . . , vn} v∈ V v {v1, . . . , v_n} linear combination. v = c₁v₁+··· + cnv_n,

⟨v,vi⟩ = c1⟨v1, vi⟩ + ··· + cn⟨vn, vi⟩ = ci⟨vi, vi⟩,

c_i= ⟨v,vi⟩

⟨vi, v_i⟩.

V finite dimensional , Gram-Schmidt orthogonalization process

V orthogonal basis ( orthonormal basis). process.

w₁∈ V \ {OV} v₁= w1. w₂∈ V \ Span({w1}), v₂= w₂−⟨w2, v₁⟩

⟨v1, v₁⟩v₁.

⟨v1, v2⟩ = 0 Span({v1, v2}) = Span({w1, w2}). Span({v1, v2}) = V, {v1, v₂} V orthogonal basis. 再 w₃∈ V \ Span({w1, w₂}),

v₃= w₃−

(⟨w3, v₁⟩

⟨v1, v₁⟩v₁+⟨w3, v₂⟩

⟨v2, v₂⟩v₂ )

.

⟨v1, v₃⟩ = ⟨v2, v₃⟩ = 0 Span({v1, v₂, v₃}) = Span({w1, w₂, w₃}). , w_i∈ V \ Span({w1, . . . , wi−1}),

v_i= w_i−

(⟨wi, v₁⟩

⟨v1, v1⟩v₁+··· + ⟨wi, v_i−1⟩

⟨vi−1, vi−1⟩v_i−1 )

.

⟨v1, v_i⟩ = ··· = ⟨vi−1, v_i⟩ = 0 Span({v1, . . . , v_i}) = Span({w1, . . . , w_i}). V

finite dimensional, . {v1, . . . , v_n} V orthogonal

basis. 再 , 前 v_i ∥vi∥⁻¹ orthonormal basis.

, V basis{w1, . . . , wn}, w_i∈ V \Span({w1, . . . , wi−1}),

process, V orthogonal basis.

W V subspace , W^′ V subspace V = W⊕W^′,

W^′ . inner product space , W^′

. .

Definition 5.1.10. V inner product space, S V nonempty subset.

S^⊥={v ∈ V | ⟨v,w⟩ = 0, ∀w ∈ S}.

S^⊥ the orthogonal complement of S in V . Question 5.4. {OV}^⊥? V^⊥?

S^⊥ 性 .

Lemma 5.1.11. V inner product space.

(1) S V nonempty subset, S^⊥ V subspace.

(2) S1, S2 V nonempty subsets S1⊆ S2, S^⊥₂ ⊆ S^⊥₁. (3) S V nonempty subset, S^⊥= Span(S)^⊥.

(4) W V subspace, W∩W^⊥={OV}.

Proof. V inner product space over F ( F =C F =R).

(1) O_V ∈ S^⊥. v₁, v₂∈ S^⊥, r, s∈ F ⟨rv1+ sv₂, w⟩ = r⟨v1, w⟩ + s⟨v2, w⟩ = 0, ∀w ∈ S. rv1+ sv₂∈ S^⊥. S^⊥ V subspace.

(2) v∈ S^⊥₂, w∈ S2 ⟨v,w⟩ = 0. w∈ S1 S₁⊆ S2, w∈ S2, v∈ S^⊥₁, S^⊥₂ ⊆ S^⊥₁.

(3) S⊆ Span(S), (2) Span(S)^⊥ ⊆ S^⊥. , v∈ S^⊥,

w∈ Span(S), c₁, . . . , c_n∈ F w₁. . . , w_n∈ S w = c₁w₁+···+cnw_n,

⟨w,v⟩ = c1⟨w1, v⟩+···+cn⟨wn, v⟩ = 0. v∈ Span(S)^⊥, S^⊥⊆ Span(S)^⊥, S^⊥= Span(S)^⊥.

(4) v∈ W ∩W^⊥, v⊥ v, ⟨v,v⟩ = 0. inner product 性 v = O_V.

 Question 5.5. Lemma 5.1.11 (1) S V subspace, S^⊥ V

subspace. (4) W V subspace?

Question 5.6. W1,W2 V subspaces, (W1+W2)^⊥= W₁^⊥∩W₂^⊥.

W V subspace Gram-Schmidt process W orthogonal

basis S ={w1, . . . , wk}. v∈ V

˜v = ⟨v,w1⟩

⟨w1, w1⟩w₁+··· + ⟨v,wk⟩

⟨wk, w_k⟩w_k,

˜v∈ W w_i ⟨v − ˜v,wi⟩ = 0. Lemma 5.1.11 (3)

v− ˜v ∈ S^⊥= Span(S)^⊥= W^⊥. 性 .

5.1. Inner Product Spaces 111

Proposition 5.1.12. V inner product space W V finite dimensional

subspace. v∈ V, ˜v∈ W 性 .

(1) v− ˜v ∈ W^⊥

(2) w∈ W \ {˜v}, ∥v − ˜v∥ < ∥v − w∥.

(3) ∥˜v∥ ≤ ∥v∥.

Proof. W orthogonal basis S ={w1, . . . , wk}, v∈ V, 前 v− ˜v ∈ W^⊥. (1).

w∈ W, ˜v− w ∈ W,

⟨v − w,v − w⟩ = ⟨v − ˜v + ˜v − w,v − ˜v + ˜v − w⟩ = ⟨v − ˜v,v − ˜v⟩ + ⟨˜v − w, ˜v − w⟩.

∥v − w∥ ≥ ∥v − ˜v∥ ⟨˜v − w, ˜v − w⟩ = 0, w = ˜v. (2).

v− ˜v ∈ W^⊥ ˜v∈ W,

⟨v,v⟩ = ⟨v − ˜v + ˜v,v − ˜v + ˜v⟩ = ⟨v − ˜v,v − ˜v⟩ + ⟨˜v, ˜v⟩.

∥˜v∥ ≤ ∥v∥. (3). 

Proposition 5.1.12 (2) ˜v W v , ˜v ,

W orthogonal basis . proj_W(v) ˜v the

projection of v on W . V finite dimensional, W V finite

dimensional subspace, Proposition 5.1.12 . W finite dimensional,

Proposition 5.1.12 .

Question 5.7. Proposition 5.1.12 ∥˜v∥ = ∥v∥ v∈ W.

v∈ V, v = v− projW(v) + proj_W(v). v− projW(v)∈ W^⊥ proj_W(v)∈ W, V = W +W^⊥. Lemma 5.1.11 (4), W∩W^⊥={OV},

.

Theorem 5.1.13. V inner product space W V finite dimensional subspace.

V = W⊕W^⊥.

V finite dimensional, subspace finite dimensional,

Theorem 5.1.13 V subspace . W^⊥ subspace,

V = W^⊥⊕ (W^⊥)^⊥. W = (W^⊥)^⊥?

Corollary 5.1.14. V inner product space W V finite dimensional subspace.

(W^⊥)^⊥= W.

Proof. w∈ W, v∈ W^⊥, ⟨w,v⟩ = 0, w∈ (W^⊥)^⊥. W⊆ (W^⊥)^⊥. v∈ (W^⊥)^⊥, Theorem 5.1.13, v v = w + w^′, w∈ W w^′∈ W^⊥. v∈ (W^⊥)^⊥, ⟨v,w^′⟩ = 0,

0 =⟨v,w^′⟩ = ⟨w,w^′⟩ + ⟨w^′, w^′⟩ = ⟨w^′, w^′⟩.

w^′= OV, v = w∈ W, (W^⊥)^⊥⊆ W. 

Question 5.8. inner product space subspace W ( finite dimensional) W^⊥=(

(W^⊥)^⊥)_⊥ .

Question 5.9. V finite dimensional inner product space, S V subset.

(S^⊥)^⊥ ? W₁,W₂ V subspace, (W₁∩W2)^⊥= W₁^⊥+W₂^⊥ V subsets S, S^′, v∈ S,v^′∈ S^′ ⟨v,v^′⟩ = 0, S⊥ S^′ . , W,W^′ V subspaces W ⊥ W^′, W^′⊆ W^⊥, Lemma 5.1.11

W∩W^′={OV}. W1, . . . ,W_k V subspaces V = W1+··· +Wk,

i̸= j, W_i⊥ Wj, V W₁, . . . ,W_k direct sum. V orthogonal direct sum direct sum

V = W₁
···
Wk

. W V finite dimensional subspace, Theorem 5.1.13 V = W
W^⊥.

5.2. Dual Spaces

Dual space inner product space 性, inner

product space 性 dual space . dual

space.

Definition 5.2.1. V vector space over F, f : V → F F-linear transformation, f linear functional on V . linear functional on V

vector space over F, V dual space, V^∗ .

Question 5.10. n× n determinant 數 det : M_n(F)→ F trace 數 tr : Mn(F)→ F. linear functional on Mn(F)?

V basis, {v1, . . . , v_n} w₁, . . . , w_n∈ W,

linear transformation T : V → W T (vi) = wi,∀i = 1,...,n. i = 1, . . . , n v^∗_i : V → F linear function on V ,

v^∗_i(vj) =

{ 1, if j = i;

0, if j̸= i.

性 .

5.2. Dual Spaces 113

Theorem 5.2.2. {v1, . . . , v_n} V basis, {v^∗₁, . . . , v^∗_n} V^∗ basis.

, dim(V ) = dim(V^∗).

Proof. Span({v^∗₁, . . . , v^∗_n}) = V^∗. f ∈ V^∗, c1, . . . , cn∈ F f = c₁v^∗₁+···+cnv^∗_n. 前 linear transformation basis 性 ,

c1, . . . , c_n∈ F f c1v^∗₁+···+cnv^∗_n v_i . v_i,

(c1v^∗₁+··· + cnv^∗_n)(vi) = c1v^∗₁(vi) +··· + cnv^∗_n(vi) = civ^∗_i(vi) = ci. (5.3) c_i= f (v_i), f = c₁v^∗₁+··· + cnv^∗_n.

{v^∗₁, . . . , v^∗_n} linearly independent. c₁v^∗₁+··· + cnv^∗_n = O zero mapping. v_i, (c₁v^∗₁+··· + cnv^∗_n)(v_i) = 0, (5.3) ci = 0,

∀i = 1,...,n. 

V basis{v1, . . . , v_n}, {v^∗₁, . . . , v^∗_n} {v1, . . . , v_n} dual basis.

Question 5.11. {v1, . . . , vn} V basis, v∈ V, v^∗_i(v) ?

V^∗ F-space, V^∗ dual space ? (V^∗)^∗( V double

dual space). (V^∗)^∗ linear functional on V^∗. σ ∈ (V^∗)^∗,

σ : V^∗→ F linear transformation f∈ V^∗ F . , v∈ V,

ˆv : V^∗→ F f ∈ V^∗, ˆv( f ) = f (v). ˆv∈ (V^∗)^∗, ˆv linear functional, f , g∈ V^∗ r, s∈ F,

ˆv(r f + sg) = (r f + sg)(v) = r f (v) + sg(v) = r ˆv( f ) + sˆv(g).

Theorem 5.2.2 V finite dimensional , dim(V ) = dim(V^∗),

dim(V^∗) = dim((V^∗)^∗). dim(V ) = dim((V^∗)^∗). V (V^∗)^∗ isomorphic,

ˆv V (V^∗)^∗ isomorphism, V (V^∗)^∗

canonical map

Proposition 5.2.3. τ : V → (V^∗)^∗ τ(v) = ˆv, ∀v ∈ V. τ one-to-one linear transformation. V finite dimensional , τ isomorphism.

Proof. τ linear transformation. v, w∈ V r, s∈ F τ(rv + sw)( f ) = f (rv + sw) = r f (v) + s f (w) = (rτ(v) + sτ(w))( f ), ∀ f ∈ V^∗,

τ(rv + sw) rτ(v) + sτ(w) V^∗ 數, τ(rv + sw) =

rτ(v) + sτ(w).

τ one-to-one, Ker(τ) = OV. v∈ Ker(τ), τ(v) = ˆv V^∗ zero mapping. f∈ V^∗ 0 = ˆv( f ) = f (v). v̸= OV,

f∈ V^∗ f (v)̸= 0. v = O_V.

V finite dimensional, τ :V → (V^∗)^∗ one-to-one dim(V ) = dim((V^∗)^∗),

τ onto, τ isomorphism. 

dual space orthogonal complement , .

Definition 5.2.4. V vector space, S V nonempty subset.

S⁰={ f ∈ V^∗| f (v) = 0, ∀v ∈ S}.

S⁰ the annihilator of S.

Question 5.12. {OV}⁰? V⁰?

S⁰ Lemma 5.1.11 性 .

Lemma 5.2.5. V vector space over F.

(1) S V nonempty subset, S⁰ V^∗ subspace.

(2) S₁, S₂ V nonempty subsets S₁⊆ S2, S⁰₂⊆ S⁰₁. (3) S V nonempty subset, S⁰= Span(S)⁰.

Proof.

(1) V^∗ zero mapping O S⁰. f , g∈ S⁰, r, s∈ F

r f + sg(v) = r f (v) + sg(v) = 0, ∀v ∈ S. r f + sg∈ S⁰. S⁰ V^∗ subspace.

(2) f∈ S⁰₂, w∈ S2 f (w) = 0. w∈ S1 S₁⊆ S2, w∈ S2, f∈ S⁰₁, S⁰₂⊆ S⁰₁.

(3) S⊆ Span(S), (2) Span(S)⁰⊆ S⁰. , f ∈ S⁰, w∈

Span(S), c₁, . . . , c_n∈ F w₁. . . , w_n∈ S w = c₁w₁+··· + cnw_n, f (w) = c1f (w1) +··· + cnf (wn) = 0. f ∈ Span(S)⁰, S⁰⊆ Span(S)⁰, S⁰= Span(S)⁰.

 V finite dimensional inner product space, W V subspace, Theorem

5.1.13 dim(W^⊥) = dim(V )− dim(W). annihilator 性 .

Proposition 5.2.6. V = W⊕U. isomorphismϕ : U^∗→ W⁰. V finite dimensional, subspace W dim(W⁰) = dim(V )− dim(W).

Proof. direct sum 性 , v∈ V, w∈ W,u ∈ U v = w + u.

f ∈ U^∗, ϕ( f ) : V → F ϕ( f )(w + u) = f (u). f linear functional

ϕ( f ) linear functional, ϕ( f ) ∈ V^∗. w∈ W,

w = w + O_U, ϕ( f )(w) = f (OU) = 0, ϕ( f ) ∈ W⁰. ϕ : U^∗→ W⁰

5.2. Dual Spaces 115

well-defined function. ϕ linear transformation. f , g∈ U^∗ f ̸= g, u∈ U f (u)̸= g(u), ϕ( f )(u) ̸= ϕ(g)(u), ϕ( f ) ̸= ϕ(g), ϕ

one-to-one. ϕ onto f ∈ W⁰, f (v) = f (w + u) = f (u),

w∈ W,u ∈ U v = w + u. f|U∈ U^∗, ϕ( f |U) = f , ϕ onto.

V finite dimensional, subspace W , W basis

{w1, . . . , wk} 大 V basis {w1, . . . , wk, u1, . . . , ul}. V = W⊕U, U = Span({u1, . . . , u_l}). Theorem 5.2.2 ,

dim(W⁰) = dim(U^∗) = dim(U ) = dim(V )− dim(W).

 Question 5.13. W1,W2 V subspaces, (W1+ W2)⁰= W₁⁰∩W₂⁰. V finite dimensional, (W₁∩W2)⁰= W₁⁰+W₂⁰.

W⁰ V^∗ subspace, W⁰ annihilator, (W⁰)⁰. Corollary 5.1.14 性 .

Corollary 5.2.7. V finite dimensional vector space, W V subspace.

canonical map τ : V → (V^∗)^∗ τ(v) = ˆv, ∀v ∈ V, τ(W) = (W⁰)⁰.

Proof. w∈W, f∈W⁰, w( f ) = f (w) =ˆ 0, τ(w) = ˆw ∈ (W⁰)⁰. τ(W) ⊆ (W⁰)⁰. Proposition 5.2.3, τ : V → (V^∗)^∗ one-to-one, dim(τ(W)) = dim(W ). Proposition 5.2.6

dim((W⁰)⁰) = dim(V^∗)− dim(W⁰) = dim(V )− (dim(V) − dim(W)) = dim(W).

dim(τ(W)) = dim((W⁰)⁰), τ(W) = (W⁰)⁰. 

探 dual space inner product space . .

Definition 5.2.8. V,W vector spaces over F, F C R. T : V → W v, v^′∈ V, r ∈ F T (v + v^′) = T (v) + T (v^′) T (rv) = rT (v), T

conjugate transformation. T one-to-one and onto, conjugate isomorphism.

F =R , conjugate transformation linear transformation. F =C conjugate transformation linear transformation ,

conjugate transformation 性 .

Lemma 5.2.9. V,W,U vector spaces overC. T1: V → W, T2: W → U.

(1) T1, T2 linear transformation conjugate transformation, T₂◦ T1: V → U conjugate transformation.

(2) T₁, T₂ conjugate transformation, T₂◦T1: V→ U linear transformation.

(3) T₁ conjugate isomorphism, T₁⁻¹: W → V conjugate isomorphism.

Proof.

(1) T1 conjugate transformation T2 linear transformation ,

, . v, v^′∈ V r, s∈ C T₂◦ T1(rv + sv^′) =

rT2◦ T1(v) + sT2◦ T1(v^′).

T₂◦ T1(rv + sv^′) = T₂(rT₁(v) + sT₁(v^′)) = rT₂(T₁(v)) + sT₂(T₁(v^′)) = rT₂◦ T1(v) + sT₂◦ T1(v^′).

T₂◦ T1 conjugate transformation.

(2) T₁, T₂ conjugate transformation, v, v^′∈ V r, s∈ C T2◦ T1(rv + sv^′) = rT2◦ T1(v) + sT2◦ T1(v^′).

T₂◦ T1(rv + sv^′) = T₂(rT₁(v) + sT₁(v^′)) = (r)T₂(T₁(v)) + (s)T₂(T₁(v^′)) = rT₂◦ T1(v) + sT₂◦ T1(v^′).

T₂◦ T1: V → U linear transformation.

(3) T₁ conjugate isomorphism, T₁ one-to-one and onto, T₁⁻¹: W → V

T1(T₁⁻¹(w)) = w,∀w ∈ W. w, w^′∈ W r, s∈ C, T1(T₁⁻¹(rw + sw^′)) = rw + sw^′

T1(rT⁻¹(w) + sT⁻¹(w^′)) = (r)T₁(T⁻¹(w)) + (s)T₁(T⁻¹(w^′)) = rw + sw^′, T1(T⁻¹(rw + sw^′)) = T1(rT⁻¹(w) + sT⁻¹(w^′)). T1 one-to-one,

T⁻¹(rw + sw^′) = rT⁻¹(w) + sT⁻¹(w^′).

T₁⁻¹ conjugate isomorphism. 

V inner product space over F, v∈ V, 數 ρ(v) : V → F w7→ ⟨w,v⟩, ( ρ(v) = ⟨·,v⟩) ρ(v) linear functional on V , ρ(v) ∈ V^∗. ρ

V V^∗ mapping. v, v^′∈ V,

ρ(v + v^′)(w) =⟨w,v + v^′⟩ = ⟨w,v⟩ + ⟨w,v^′⟩ =ρ(v)(w) + ρ(v^′)(w), ∀w ∈ V.

ρ(v + v^′) =ρ(v) + ρ(v^′) in V^∗. r∈ F,

ρ(rv)(w) = ⟨w,rv⟩ = r⟨w,v⟩ = rρ(v)(w), ∀w ∈ V.

ρ(rv) = rρ(v) in V^∗. .

Proposition 5.2.10. V finite dimensional inner product space. ρ : V → V^∗ ρ(v) = ⟨·,v⟩, ρ conjugate isomorphism.

Proof. ρ conjugate transformation, ρ one-to-one and onto.

ρ one-to-one. v, v^′ ∈ V ρ(v) = ρ(v^′), ⟨w,v⟩ = ⟨w,v^′⟩,

∀w ∈ V. Corollary 5.1.5 v = v^′, ρ one-to-one.

5.3. Transpose and Adjoint 117

ρ onto, V orthonormal basis{v1, . . . , v_n}. f ∈ V^∗, v = f (v₁)v₁+··· + f (vn)v_n.

ρ(v)(vi) =⟨vi, f (v₁)v₁+··· + f (vn)v_n⟩ = ⟨vi, f (v_i)v_i⟩ = f (vi), ∀i ∈ {1,...,n}.

f ρ(v) {v1, . . . , vn} basis , f =ρ(v). ρ

onto. 

ρ conjugate isomorphism, Lemma 5.2.9 ρ⁻¹: V^∗→ V conjugate isomorphism.

Question 5.14. Proposition 5.2.10 ρ : V → V^∗ one-to-one dim(V ) = dim(V^∗) ρ : V → V^∗ onto ?

Example 5.2.11. Rⁿ standard inner product, {e1, . . . , e_n} Rⁿ standard basis. v = x₁e₁+···+xne_n∈ Rⁿ, ρ(v)(ei) =⟨ei, v⟩ = xi. ρ(v) = f ∈ (Rⁿ)^∗,

f (e_i) = x_i,∀i ∈ {1,...,n}. , f∈ (Rⁿ)^∗, ρ⁻¹( f ) = f (e₁)e₁+··· + f (en)e_n. Cⁿ standard inner product, {e1, . . . , e_n} Cⁿ standard basis.

v = z₁e₁+··· + zne_n∈ Cⁿ, ρ(v)(ei) =⟨ei, v⟩ = zi. ρ(v) = f ∈ (Cⁿ)^∗, f (e_i) = z_i,∀i ∈ {1,...,n}. , f∈ (Cⁿ)^∗, ρ⁻¹( f ) = f (e₁)e₁+··· + f (en)e_n. 5.3. Transpose and Adjoint

linear operator, transpose adjoint 性 ,

linear transformation . linear

transformation. linear transformation T : V → W, dual spaces W^∗,V^∗

T transpose, V,W inner product space, T adjoint.

探 transpose adjoint .

探 linear transformation T : V→ W transpose, V,W vector

space over F. 再 T transpose V,W inner product space

finite dimensional. f∈ W^∗, T, f linear transformation, f◦ T : V → F linear transformation. f◦ T ∈ V^∗.

mapping T^t : W^∗→ V^∗, T^t( f ) = f◦ T, ∀ f ∈ W^∗. , f , g∈ W^∗, r, s∈ F

T^t(r f + sg) = (r f + sg)◦ T = r( f ◦ T) + s(g ◦ T) = rT^t( f ) + sT^t(g),

T^t: W^∗→ V^∗ linear transformation. T^t T transpose. T T^t , v∈ V, f ∈ W^∗

f (T (v)) = T^t( f )(v).

transpose 性 .

Proposition 5.3.1. V,W,U vector space over F.

(1) r, s∈ F T₁: V → W, T2: V → W linear transformations, (rT₁+ sT₂)^t= rT₁^t+ sT₂^t.

(2) T1: V → W, T2: W→ U linear transformations, (T2◦ T1)^t= T₁^t◦ T2^t.

(3) (id_V)^t= id_V∗, , T : V→ W isomorphism, T^t: W^∗→ V^∗ isomor- phism

(T^t)⁻¹= (T⁻¹)^t. Proof.

(1) f ∈ W^∗, f linear,

(rT1+ sT2)^t( f ) = f◦ (rT1+ sT2) = r( f◦ T1) + s( f◦ T2) = rT₁^t( f ) + sT₂^t( f ) = (rT₁^t+ sT₂^t)( f ).

(rT₁+ sT₂)^t= rT₁^t+ sT₂^t.

(2) T2◦ T1 V U linear transformation, (T2◦ T1)^t U^∗ V^∗ linear transformation. f∈ U^∗,

(T₂◦ T1)^t( f ) = f◦ (T2◦ T1) = ( f◦ T2)◦ T1= T₁^t( f◦ T2) = T₁^t(T₂^t( f )) = T₁^t◦ T₂^t( f ).

(T2◦ T1)^t= T₁^t◦ T₂^t.

(3) f ∈ V^∗, (idV)^t( f ) = f◦ idV = f , (idV)^t= idV^∗. T : V → W isomorphism, T⁻¹◦ T = idV T◦ T⁻¹= id_W. (2)

id_V∗ = (id_V)^t= T^t◦ (T⁻¹)^t, id_W∗= (id_W)^t= (T⁻¹)^t◦ T^t,

T^t isomorphism (T^t)⁻¹= (T⁻¹)^t. 

linear transformation kernel image transpose

kernel image . vector space , 探

finite dimensional .

Proposition 5.3.2. V,W finite dimensional vector space, T : V → W linear transformation, T^t: W^∗→ V^∗ transpose.

Ker(T^t) = Im(T )⁰, Im(T^t) = Ker(T )⁰.

Proof. f∈ Ker(T^t), T^t( f ) = f◦ T = O in V^∗. v∈ V, T^t( f )(v) = f (T (v)) = 0. f∈ {T(v) | v ∈ V}⁰= Im(T )⁰. , f ∈ Im(T)⁰, v∈ V, f (T (v)) = 0, T^t( f ) = O in V^∗, f∈ Ker(T^t).

, f∈ Im(T^t), g∈ W^∗ f = T^t(g) = g◦ T. v∈ Ker(T), f (v) = g(T (v)) = g(OW) = 0. f ∈ Ker(T)⁰, Im(T^t)⊆ Ker(T)⁰. T^t: W^∗→ V^∗ linear transformation, Theorem 5.2.2

dim(Im(T^t)) = dim(W^∗)− dim(Ker(T^t)) = dim(W )− dim(Ker(T^t)).

5.3. Transpose and Adjoint 119

再 Ker(T^t) = Im(T )⁰ Proposition 5.2.6

dim(Ker(T^t)) = dim(Im(T )⁰) = dim(W )− dim(Im(T)),

dim(Im(T^t)) = dim(Im(T )).

T : V → W linear transformation, dim(Ker(T )) = dim(V )− dim(Im(T)), 再 Proposition 5.2.6

dim(Ker(T )⁰) = dim(V )− dim(Ker(T)) = dim(Im(T)).

dim(Im(T^t)) = dim(Ker(T )⁰), Im(T^t) = Ker(T )⁰.  Question 5.15. Proposition 5.3.2 dim(Im(T^t)) = dim(Im(T )).

dim(Ker(T^t)) = dim(Ker(T ))?

T^t: W^∗→ V^∗ linear transformation, T^t transpose, (T^t)^t: (V^∗)^∗→ (W^∗)^∗. V finite dimensional, isomorphism τV : V → (V^∗)^∗, v∈ V, τV(v) = ˆv ˆv( f ) = f (v), ∀ f ∈ V^∗.

:

V −−−−−−→^T W



y^τV y^τW (V^∗)^{∗ (T}

t)^t

−−−−−−→ (W^∗)^∗

v∈ V, (T^t)^t(τV(v)) = ˆv◦ T^t∈ (W^∗)^∗. f ∈ W^∗,

(T^t)^t(τV(v))( f ) = ˆv◦ T^t( f ) = ˆv( f◦ T) = f ◦ T(v) = f (T(v)).

T (v)∈ W, τW(T (v)) = dT (v)∈ (W^∗)^∗. f ∈ W^∗, τW(T (v))( f ) = dT (v)( f ) = f (T (v)).

(T^t)^t(τV(v)) =τW(T (v)), ∀v ∈ V,

(T^t)^t◦τV =τW◦ T.

commutative diagram. 再 τV isomorphism, τV⁻¹: (V^∗)^∗→ V ( isomorphism). .

Proposition 5.3.3. V,W finite dimensional vector space, T : V → W linear transformation. τV τW V, (V^∗)^∗ W, (W^∗)^∗ canonical map.

(T^t)^t=τW◦ T ◦τV⁻¹.

Linear transformation T : V → W T transpose T^t: W^∗→ V^∗ V,W ordered basis dual basis representative matrix .

Proposition 5.3.4. β = (v1, . . . , v_n),γ = (w1, . . . , w_m) V,W ordered basis β^∗= (v^∗₁, . . . , v^∗_n),γ^∗= (w^∗₁, . . . , w^∗_m) dual basis V^∗,W^∗ ordered basis. Linear transformation T : V → W,

β^∗[T^t]_γ^∗=_γ[T ]^t_β.

Proof. T^t(w^∗_i) = w^∗_i ◦ T ∈ V^∗, w^∗_i ◦ T = c1,iv^∗₁+··· + cn,iv^∗_n, c_j,i= w^∗_i ◦ T(vj) = w^∗_i(T (v_j)).

T (vj) = d1, jw₁+··· + dm, jw_m, cj,i= di, j. cj,i β^∗[T^t]_γ∗ ( j, i)-th entry, d_{i, j γ}[T ]_β (i, j)-th entry, _β∗[T^t]_γ∗=_γ[T ]^t_β. 

Proposition 5.3.4, linear transformation transpose matrix transpose . Proposition 5.3.1, 5.3.2, 5.3.3 matrix transpose

性 .

Question 5.16. β = (v1, . . . , v_n),γ = (w1, . . . , w_m) V,W ordered basis β = ( bˆ v₁, . . . ,vb_n), ˆγ = (cw₁, . . . ,wc_m) (V^∗)^∗, (W^∗)^∗ ordered basis, vb_i=τV(vi),cw_j= τW(w_j) (τV : V → (V^∗)^∗, τW : W → (W^∗)^∗ canonical maps.) Proposition 5.3.3, 5.3.4

(_γ[T ]^t_β)^t=_γˆ[(T^t)^t]_βˆ =_γ[T ]_β.

探 linear transformation T : V → W adjoint. 再 adjoint V,W finite dimensional inner product spaces. ,

ρV : V→ V^∗,ρW: W→ W^∗ ρV(v) =⟨·,v⟩,∀v ∈ V ρW(w) =⟨·,w⟩,∀w ∈ W ρV,ρW conjugate isomorphism. T adjoint T^∗: W→ V

T^∗=ρV⁻¹◦ T^t◦ρW.

言 w∈ W,

T^∗(w) =ρV⁻¹◦ρW(w)◦ T =ρV⁻¹(⟨T(·),w⟩).

Lemma 5.2.9 (1) T^t◦ρW : W → V^∗ conjugate transformation, 再 Lemma 5.2.9 (2) T^∗: W→ V linear transformation. commutative diagram.

Vy^ρV ←−−−−−− W^T∗ y^ρW

V^∗ ←−−−−−− W^{Tt ∗}

Theorem 5.3.5. V,W finite dimensional inner product space, T : V → W linear transformation. T transpose T^∗: W → V, linear transformation

⟨T(v),w⟩ = ⟨v,T^∗(w)⟩, ∀v ∈ V,w ∈ W.