Linear Operator

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 (

).

, ,

代. , .

, . , 性

, . , .

, 大 . 大

, . ,

.

v

前言 v

Chapter 1. Vector Spaces 1

§1.1. Definition and Basic Properties 1

§1.2. Subspaces 4

§1.3. Spanning Sets 7

§1.4. Linear Independence 9

§1.5. Basis and Dimension 11

§1.6. Direct Sum and Quotient Space 17

Chapter 2. Linear Transformations 23

§2.1. Definition and Basic Properties 23

§2.2. Image and Kernel 25

§2.3. Isomorphism 29

§2.4. The Matrix Connection 32

Chapter 3. Linear Operator 41

§3.1. Basic Concept 41

§3.2. Characteristic Polynomial 44

§3.3. Minimal Polynomial 49

§3.4. Internal Direct Sum 53

§3.5. Primary Decomposition 56

Chapter 4. Form Reduction 65

§4.1. Diagonal From 65

vii

§4.2. Triangular Form 72

§4.3. Jordan Form 79

§4.4. Rational Form 88

§4.5. Classical Form 95

Chapter 5. Operators on Inner Product Spaces 105

§5.1. Inner Product Spaces 105

§5.2. Dual Spaces 112

§5.3. Transpose and Adjoint 117

§5.4. The Adjoint of Linear Operators 125

§5.5. Normal Operators 130

§5.6. The Spectral Theorem 135

Vector Spaces

linear algebra 探 “Vector Space”. 探

vector space basis dimension.

1.1. Definition and Basic Properties

vector space , (vector).

性 . , “+”

vector space . vector space

V ( V ), 再 “+”. 性,

v, w∈ V v + w∈ V. 性

VS1: u, v∈ V, u + v = v + u.

VS2: u, v, w∈ V, (u + v) + w = u + (v + w).

VS3: O∈ V u∈ V O + u = u.

VS4: u∈ V u^′∈ V u + u^′= O.

大 , V + , abelian group. vector space

group, filed “ ” (action), scaler

multiplication. 性. vector space V ,

field F, r∈ F v∈ V, r v , ( rv),

V .

+ ,

性

VS5: r, s∈ F u∈ V, r(su) = (rs)u.

VS6: r, s∈ F u∈ V, (r + s)u = ru + su.

VS7: r∈ F u, v∈ V r(u + v) = ru + rv.

1

VS8: u∈ V, 1u = u.

性 ? VS1∼ VS8 8 性 , vector

space 數 . 8 性 ,

性 . , .

V , field F .

V O , F 0 . V

F , ( V = F), + .

V = F, V F ,

. 再 , vector space abelian group V

filed F. V , F 代數 V , F .

vector space .

V over F vector space. V F-space.

Example 1.1.1. vector space. F filed.

(1) Fⁿ={(a1, . . . , a_n)| ai∈ F}. : (a₁, . . . , a_n), (b₁, . . . , b_n)∈ Fⁿ, (a1, . . . , an) + (b1, . . . , bn) = (a1+ b1, . . . , an+ bn). Fⁿ

VS1 ∼ VS4 , O = (0, . . . , 0). F

Fⁿ : r∈ F, (a1, . . . , a_n)∈ Fⁿ, r(a1, . . . , a_n) = (a₁, . . . , a_n).

VS5∼ VS8. Fⁿ over F

vector space. F over F vector space.

Mm×n(F) entry F m× n . (

), M_m×n(F) over F vector space.

(2) 數 F 數 n , : f (x) = anxⁿ+··· +

a₀, g(x) = b_nxⁿ+··· + b0, f (x) + g(x) = (a_n+ b_n)xⁿ+··· + (a0+ b₀).

數 F 數 n . (

數 數 ). 數 n

, , VS1 ∼ VS4. r∈ F,

f (x) = anxⁿ+··· + a0, r f (x) = ranxⁿ+··· + ra0, 數

F 數 n vector space over F. P_n(F)

vector space. Pn(F) O ( ).

(3) S. S F 數 F^S. r∈ F, f ,g ∈ F^S,

f + g, r f ∈ F^S f + g : s7→ f (s)+ g(s) r f : s7→ r f (s).

F^S over F vector space f∈ F^S f (s) = 0,∀s ∈ S F^S O.

Question 1.1. Example 1.1.1 (1) (3) ? (1) (2)

?

Question 1.2. V vector space over F, V^′ V subgroup F^′ F

subfield. V , F , V^′ over F vector space? V

over F^′ vector space?

vector space 性 . V F-space, V

abelian group 性 , . : VS3

O∈ V v∈ V v + O = v. O ,

. VS4 v∈ V v^′∈ V v + v^′= O, v^′

v , , −v v^′.

, V w, v∈ V v + w = v,

w = O.

VS5∼ VS8, V F , 性 .

Proposition 1.1.2. V over F vector space, 性 : (1) r∈ F, v ∈ V, rv = O r = 0 v = O.

(2) v∈ V, v + (−1)v = O. 言 ,−v = (−1)v.

Proof. (1) (⇐) r = 0, rv = O, , 0v+v = v

. VS6 VS8

0v + v = 0v + 1v = (0 + 1)v = 1v = v.

. , v = O, rv = O rO + rO = rO .

VS3 VS7

rO + rO = r(O + O) = rO.

.

(⇒) rv = O r̸= 0, F field r⁻¹∈ F r⁻¹r = 1.

VS5, VS8 前

v =1v = r⁻¹(rv) = r⁻¹O = O.

.

(2) VS8, VS6 (1) ,

v + (−1)v = 1v + (−1)v = (1 − 1)v = 0v = O.

性 −v = (−1)v.

 Question 1.3. V F-space. Proposition 1.1.2 性 ?

(1) r, r^′∈ F, u,v ∈ V r̸= r^′, u̸= O, ru + v̸= r^′u + v.

(2) r∈ F, v ∈ V, −(rv) = (−r)v = r(−v).

1.2. Subspaces

V over F vector space, U V (nonempty subset),

V , F , U over F vector space, U V subspace.

U V F-subspace over F subspace.

U≤ V U V subspace.

V subset S V subspace ?

, subspace, S abelian group, S V

subgroup. S V subgroup, 前 Question 1.2 S

F-space. S F 性. S V

F S V subspace.

Proposition 1.2.1. V over F vector space S V subset. S

V F-subspace S 性 :

(1) O∈ S.

(2) r, s∈ F,u,v ∈ S ru + sv∈ S.

Proof. (⇒) S V subspace, S , v S . subspace

S F-space, S, F 性 Proposition 1.1.2 (1) 0v = O∈ S.

r, s∈ F,u,v ∈ S, S, F 性 ru, sv∈ S, 再 S 性 ru + sv∈ S.

(⇐) (1) O∈ S S . S F-space, S

性 S, F 性, 再 VS1∼ VS8 . u, v∈ S S⊆ V,

u, v∈ V 再 V F-space, 1u = u, 1v = v. r = 1, s = 1 (2) u + v =1u + 1v∈ S,

S 性. r∈ F,u ∈ S, O∈ S, s = 1, v = O (2) ru + 1O∈ S.

u, O V , V F-space ru + 1O = ru, S, F

性. VS1∼ VS8 S . S⊆ V, VS1, VS2 VS5∼ VS8

V , S . VS3, (1) . VS4 前 S, F

性 Proposition 1.1.2 v∈ S, −v = (−1)v ∈ S. S V

F-subspace. 

Question 1.4. u∈ S r = 1, s =−1 v = u, Proposition 1.2.1 (2)

O =1u + (−1)u ∈ S. (1) ?

Question 1.5. V F-space, S V . S V

V , F , S V F-subspace.

abelian group abelian group

subgroup. vector space 前 S V subgroup ?

Proposition 1.2.1, vector space sub-

space. vector space, VS1∼ VS8 ,

vector space, .

Example 1.2.2. Example 1.1.1 vector space subspace.

(1) c1, . . . , cn ∈ F. E = {(a1, . . . , an)∈ Fⁿ| c1a1+··· + cnan = 0}, Fⁿ

subspace. E c₁x₁+··· + cnx_n= 0 . Fⁿ c₁x₁+

··· + cnxn= b Fⁿ hyperplane. b = 0

hyperplane F-space.

(2) Pn(F) 數 n− 1 , Pn−1(F), sub-

space. λ ∈ F, Λ = { f (x) ∈ Pn(F)| f (λ) = 0} P_n(F) subspace.

Pn(F) 數 0 ( λ = 0) subspace.

(3) T ⊆ S, F^S { f ∈ F^S| f (t) = 0, ∀t ∈ T} subspace.

over F vector space V , V {O} subspace.

subspace V trivial subspace. V F-subspace,

subspace F-subspace. .

Proposition 1.2.3. V vector space over F U,W V subspace, U∩W V subspace.

Proof. Proposition 1.2.1 . U,W V subspace,

O∈U O∈W, O∈U ∩W. r, s∈ F u, v∈U ∩W, u, v∈U ru + sv∈ U.

ru + sv∈ W ru + sv∈ U ∩W. 

Question 1.6. V vector space over F. I index set, i∈ I, Vi

V subspace. ^∩_i∈IV_i V subspace?

V,W V F-subspace, U∪W V F-subspace.

. R² L₁={(r,0) | r ∈ R}, L2={(0,s) | s ∈ R} R² subspace. (1, 0)∈ L1, (0, 1)∈ L2 (1, 0), (0, 1)∈ L1∪L2, (1, 1) = (1, 0) + (0, 1)̸∈

L₁∪ L2, L₁∪ L2 R² subspace. F infinite field , .

Theorem 1.2.4. F infinite field V F-space. V1, . . . ,Vn V nontrivial F-subspaces, V₁∪ ··· ∪Vn̸= V.

Proof. , V = V1∪ ··· ∪Vn. V1⊆ V2∪ ··· ∪Vn, V1

V = V2∪ ··· ∪Vn. 性 V1* V2∪ ··· ∪Vn. u∈ V1\V2∪ ··· ∪Vn

( u∈ V1 u̸∈ V2∪ ··· ∪Vn). V₁( V, v∈ V \V1. S ={ru + v | r ∈ F}.

r̸= r^′ ru + v̸= r^′u + v ( (r− r^′)u = O, u = O ).

F infinite field S .

S Vi . ru + v∈ V1, ru∈ V1,

V1 F-subspace v∈ V1 v , S∩V1= /0. , 2≤ i ≤ n,

r̸= r^′ ru + v∈ Vi r^′u + v∈ Vi, V_i F-subspace (r− r^′)u∈ Vi, 再

u∈ Vi⊆ V2∪ ··· ∪Vn. u , S Vi .

S∩ (V1∪ ··· ∪Vn) n− 1 . V F-space, u, v∈ V S⊆ V. 言 , V = V1∪ ··· ∪Vn S∩ (V1∪ ··· ∪Vn) = S∩V = S

. V = V₁∪ ··· ∪Vn . 

Theorem 1.2.4 F finite field, . F

infinite field, over F vector space subspaces

F-space ( Questions).

Question 1.7. F finite field, Theorem 1.2.4 .

Question 1.8. V over infinite field F vector space V₁, . . . ,V_n V

F-subspaces. Theorem 1.2.4 V1∪ ··· ∪Vn F-space

i∈ {1,...,n} V_j⊆ Vi,∀ j ∈ {1,...,n}.

vector space subspaces vector

space, subspaces vector space. .

Definition 1.2.5. V over F vector space V₁, . . . ,V_n V F-subspaces.

V₁+··· +Vn={v1+··· + vn| vi∈ Vi, 1≤ i ≤ n}.

, subspaces .

Question 1.9. V F-space W V subspace, W +W ?

, 性 .

Proposition 1.2.6. V over F vector space V1, . . . ,V_n V subspaces.

V1+··· +Vn V subspace.

Proof. O∈ Vi for 1≤ i ≤ n, O∈ V1+··· +Vn. , u_i, v_i∈ Vi, r, s∈ F V_i F-subspace, rui+ svi∈ Vi,

r(u₁+··· + un) + s(v₁+··· + vn) = (ru₁+ sv₁) +··· + (run+ sv_n)∈ V1+··· +Vn.

Proposition 1.2.1 V1+··· +Vn V subspace. 

Question 1.10. V over F vector space U,W V subspaces.

U∩W V U W 大 subspace. V U

W subspace?

1.3. Spanning Sets

linear combination , subspace .

Definition 1.3.1. V over F vector space S V .

v∈ V v = r₁v₁+···+rnv_n, ri∈ F v_i∈ S, v S linear combination.

Span(S) S linear combination .

, S infinite set, S linear combination S

. S linear combination v =∑u∈Sr_uu ,

ru∈ F ru 0. v =∑u∈Sruu =∑u∈Ssuu, u∈ S r_u̸= su, v S linear combination “ ”.

Question 1.11. S^′⊆ S ⊆ V, Span(S^′)⊆ Span(S) ? S

Span(S) . S Span(S)

?

S , w∈ S, 0w = O O S linear combination.

O∈ Span(S). u = r₁u₁+··· + rnu_n, v = s1v₁+··· + smv_m S linear combination ( r_i, s_j∈ F u_i, v_j∈ S), r, s∈ F,

ru + sv = r(r1u₁+··· + rnu_n) + s(s1v₁+··· + smv_m)

= (rr₁)u₁+···(rrn)u_n+ (ss₁)v₁+··· + (ssm)v_m

S linear combination. Proposition 1.2.1 .

Lemma 1.3.2. V over F vector space S V , Span(S)

V subspace.

Span(S) F-subspace the subspace spanned by S.

Span(S) V S subspace.

Proposition 1.3.3. V over F vector space S V ,

Span(S) = ^∩

S⊆W,W≤V

W,

W V S subspaces.

Proof. Lemma 1.3.2 Span(S) S subspace, Span(S)⊇ ^∩

S⊆W,W≤V

W.

W V subspace S⊆ W v∈ Span(S), v = r₁v₁+··· + rnv_n, r1∈ F, vi∈ S ⊆ W, W subspace v∈ W, Span(S)⊆ W ( Span(S)

V S subspace).

Span(S)⊆ ^∩

S⊆W,W≤V

W,

. 

S = /0, , Proposition 1.3.3 V

subspaces , {O}. S , Span(S) ={O}.

Question 1.12. V over F vector space V₁, . . . ,V_n V subspaces.

Span(Vi) = Vi. Span(V1∪ ··· ∪Vn) ?

前 Question 1.11 S Span(S) .

S Span(S) .

Corollary 1.3.4. V vector space over F S^′⊆ S ⊆ V. Span(S) = Span(S^′) S\ S^′⊆ Span(S^′).

Proof. (⇒) S\ S^′⊆ S S\ S^′⊆ Span(S). 前 Span(S) = Span(S^′) S\ S^′⊆ Span(S^′).

(⇐) S^′ ⊆ S Span(S^′)⊆ Span(S), Span(S)⊆ Span(S^′).

, S⊆ Span(S^′) . Lemma 1.3.2 Span(S^′)

subspace of V , S Span(S^′) Proposition 1.3.3 Span(S)⊆ Span(S^′).

v∈ S, v∈ S^′ v∈ S \S^′. v∈ S^′ v∈ Span(S^′); v∈ S \ S^′ v∈ Span(S^′). S⊆ Span(S^′), Span(S) = Span(S^′). 

, V F-space, S V Span(S) = V , S V

spanning set. S⊆ S^′⊆ V, S V spanning set, S^′ V

spanning set.

Example 1.3.5. Example 1.1.1 vector spaces, spanning sets.

(1) Fⁿ e_i= (0, . . . , 1, . . . , 0), 1 i , 0.

{e1, . . . , en} Fⁿ spanning set.

(2) Pn(F) anxⁿ+··· + a1x + a0 , ai ∈ F,

{1,x,...,xⁿ} P_n(F) spanning set.

(3) F^S , λ ∈ S f_λ∈ F^S, f_λ(s) =

{ 1, s =λ;

0, s̸=λ.

S finite set , { f_λ |λ ∈ S} F^S spanning set. S

infinite set , . vector space

( , “Topology”,

線性代數 ).

Question 1.13. F^S , S infinite set, Span({ f_λ |λ ∈ S}) ?

1.4. Linear Independence

linear independence , 大 ,

linear dependence . Linear independence spanning set

性 , 大 前 .

Definition 1.4.1. V over F vector space S V .

v∈ S v∈ Span({w ∈ S | w ̸= v}), S linearly dependent. , S linearly independent.

, O∈ S, S linearly dependent. O subspace .

Question 1.14. S⊆ S^′′⊆ V, S^′′ linearly independent, S linearly independent ? (or S linearly dependent, S^′′ linearly dependent)

S linearly independent, S^′′ linearly independent. linearly

independent linearly dependent.

linearly independent ?

Linear dependence .

Proposition 1.4.2. V over F vector space S V . S

linearly dependent v₁, . . . , vn∈ S r1, . . . , rn∈ F, v_i r_i 0 r₁v₁+··· + rnv_n= O.

Proof. (⇒) S linearly dependent v₁∈ S v₁∈ Span({w ∈ S | w ̸= v1}), v₂, . . . , vn∈ S v₁ r2, . . . , rn∈ F 0 v₁= r2v₂+···+rnv_n. (−1)v1+ r₂v₂+··· + rnv_n= O.

(⇐) i∈ {1,...,n}, vi∈ S r_i∈ F 0 r₁v₁+···+rnv_n= O, v₁= (−r2r⁻¹₁ )v₂+···+(−rnr⁻¹₁ )v_n∈ Span({w ∈ S | w ̸= v1}), S linearly dependent. 

Proposition 1.4.2 v₁, . . . , v_n O, n≥ 2, r₁̸= 0, r1v₁= O v₁= O.

linear independence .

Proposition 1.4.3. V over F vector space S V . S

linearly independent Span(S) S

linear combination.

Proof. (⇒) . v = O , r₁, . . . , r_n∈

F 0 v₁, . . . , vn∈ S r1v₁+··· + rnv_n = O. Proposition 1.4.2 S

linearly independent . v∈ Span(S) v̸= O ,

r₁, . . . , r_n, s₁, . . . , s_m∈ F ( r_i, s_j 0) v₁, . . . , v_n, w₁, . . . , w_m∈ S ( v₁, . . . , v_n w₁, . . . , w_m )

v = r₁v₁+··· + rnv_n= s₁w₁+··· + smw_m. : v1= w1, . . . , vk= wk v_i, wj .

O = (r₁− s1)v1+··· + (rk− sk)vk+ rk+1v_k+1+··· + rnv_n+ sk+1w_k+1+··· + smw_m. , k = n = m, ri̸= si; k < n ( r_k+1̸= 0);

k < m ( s_k+1̸= 0). Proposition 1.4.2 S linearly independent

, .

(⇐) v∈ S v∈ Span(S) v =1v v S linear

combination , v̸∈ Span({w ∈ S | w ̸= v}), S linearly

independent.

 , linearly dependent linearly independent ,

Propositions 1.4.2, 1.4.3 , . linearly

independent, linearly dependent, Proposition 1.4.2

. linearly independent, Proposition

1.4.3 性 性 .

Question 1.15. Proposition 1.4.2 Proposition 1.4.3 S ={v1, . . . , v_n} linearly independent

r1v₁+··· + rnv_n= O⇒ r1=··· = rn= 0.

Question 1.16. S linearly dependent O̸∈ S, Proposition 1.4.3

Span(S) ( ) S linear combination.

, Span(S) ( ) S linear

combination ? F infinite field , Span(S)

S linear combination.

前 Question 1.14 linearly independent set S

S linearly dependent. linearly independent set linearly independent.

Corollary 1.4.4. V vector space over F S⊆ S^′′⊆ V. S^′′ linearly independent S S^′′\S linearly independent Span(S)∩Span(S^′′\S) = {O}.

Proof. (⇒) S⊆ S^′′⊆ V S^′′ linearly independent S S^′′\S linearly independent. v∈ Span(S)∩Span(S^′′\S) v̸= O, r₁, . . . , r_n, s₁, . . . , s_m∈ F 0 v₁, . . . , v_n ∈ S, w1, . . . , w_m∈ S^′′\ S v =∑ⁿi=1r_iv_i =∑^mj=1s_jw_j.

v∈ Span(S^′′) S^′′ linear combination , Proposition 1.4.3 S^′′ linearly independent . Span(S)∩ Span(S^′′\ S) = {O}.

(⇐) , S^′′ linearly dependent, Proposition 1.4.2 r₁, . . . , r_n∈ F 0 v₁, . . . , vn ∈ S^′′ r1v₁+··· + rnv_n= O. S S^′′\ S linearly

independent, Proposition 1.4.2 v_i S S^′′\ S

. , v₁, . . . , vm∈ S v_m+1, . . . , vn ∈ S^′′\ S. 言 , r1v₁+··· + r_mv_m= (−rm+1)v_m+1+··· + (−rn)v_n∈ Span(S) ∩ Span(S^′′\ S). r₁, . . . , r_m 0 v₁, . . . , v_m linearly independent ( S linearly independent), Proposition 1.4.3 r1v₁+··· + rmv_m̸= O, Span(S)∩ Span(S^′′\ S) ̸= {O}. S^′′ linearly

independent. 

linearly independent , Example 1.3.5 spanning

set linearly independent. spanning set linearly inde-

pendent. Rⁿ {e1, e2, . . . , en, e1+ e2} Rⁿ spanning set, 再 linearly independent .

1.5. Basis and Dimension

vector space V S , S V spanning set,

S 大 , linearly independent. Basis .

.

Definition 1.5.1. V vector space over F S⊆ V. Span(S) = V S linearly independent , S V basis.

, Example 1.3.5 {e1, . . . , e_n} Fⁿ basis;

{1,x,...,xⁿ} Pn(F) basis; S finite set , { f_λ |λ ∈ S} F^S basis.

Question 1.17. 前 Proposition, S V basis

V S linear combination ?

Question 1.18. S^′( S ( S^′′ S V basis, S^′, S^′′ V basis

? Span(S^′)̸= V S^′′ linearly dependent ?

Question S V basis, spanning set

S; linearly independent set S.

, .

Proposition 1.5.2. V vector space over F S⊆ V. .

(1) S V basis.

(2) S V spanning set, S^′( S Span(S^′)̸= V.

(3) S linearly independent S^′′) S S^′′ linearly dependent.

Proof. (1)(2) , 再 (1)(3) .

((1)⇒ (2)) S basis S V spanning set. S^′( S,

Span(S^′) = V . v∈ S \S^′ v∈ V = Span(S^′)⊆ Span({w ∈ S | w ̸= v}.

S linearly independent , Span(S^′)̸= V.

((2)⇒ (1)) S spanning set, S linearly independent.

, S linearly dependent, v∈ S v∈ Span(S \ {v}).

S^′= S\ {v}. S\ S^′={v}, Corollary 1.3.4 Span(S^′) = Span(S) = V , S^′( S,

(2) 前 , S linearly independent.

((1)⇒ (3)) S basis S linearly independent. S^′′) S, S^′′ linearly independent. v∈ S^′′\ S v̸∈ Span(S). S V spanning set , S^′′ linearly dependent.

((3)⇒ (1)) S linearly independent, S V spanning

set. , Span(S)̸= V , v∈ V v̸∈ Span(S) ( Span({v}) ∪

Span(S) ={O}). S^′′= S∪ {v}. S^′′\ S = {v}, Corollary 1.4.4 S^′′ linearly

independent, S^′′) S, (3) 前 , S V spanning set.



finite set basis vector space, .

Definition 1.5.3. V vector space over F. V ={O} finite set S⊆ F V basis, V finite dimensional F-space.

V ={O} , Span( /0) ={O},

/0 {O} basis. 前 Fⁿ, Pn(F) S finite set F^S

finite dimensional vector space over F. F^′ F subfield, 前

F-space F^′-space, over filed finite

dimensional, finite dimensional F-space finite dimensional F^′-space.

Rⁿ finite dimensional R-space finite dimensional Q-space.

Definition 1.5.1 vector space basis, 探

vector space basis. 前 vector space finite

dimensional basis basis basis infinite set. 前

, vector space basis,

Zorn’s lemma. finite dimensional vector space, 再

學 Zorn’s lemma . 探 :

finite dimensional ; . ,

vector space basis. (大 ).

1.5.1. Finite Dimensional Case. V finite dimensional vector space,

finite set V basis. infinite set V basis ?

, V basis 數 .

. , finite set S, #(S) S

數.

Lemma 1.5.4. V vector space over F S⊆ V finite set Span(S) = V . S^′⊆ V #(S^′) > #(S), S^′ linearly dependent.

Proof. S ={v1, . . . , vn}, S^′={u1, . . . , um} m > n. , S^′ linearly independent.

Span(S) = V , r₁, . . . , r_n∈ F u₁= r₁v₁+··· + rnv_n. S^′ linearly independent, r1, . . . , rn 0. ( r1=··· = rn= 0 O = u₁∈ S^′, S^′

linearly independent .) 性, r₁̸= 0,

v₁= r⁻¹₁ (u1− r2u₂− ··· − rnv_n)∈ Span({u1, v2, . . . , vn}).

Corollary 1.3.4

Span({u1, v2, . . . , vn}) = Span({u1, v1, v2, . . . , vn}) = V.

u₂∈ Span({u1, v₂, . . . , v_n}) s₁, . . . , s_n∈ F u₂= s₁v₁+ s₂u₂+···+snu_n. S^′ linearly independent , s₂, . . . , s_n 0. ( s₂=··· = sn= 0, u₂= s1u₁∈ Span({u1}), S^′ linearly independent .) 性,

s₂̸= 0,

v₂= s⁻¹₂ (u₂− s1u₁− s3v₃− ··· − snv_n)∈ Span({u1, u₂, v₃, . . . , v_n}).

Corollary 1.3.4

Span({u1, u2, v3, . . . , vn}) = Span({u1, u2, v2, . . . , vn}) = V.

, v₁, . . . , v_n , u₁ 代 v₁; u₂ 代 v₂, ...

. 數學 , k < n Span({u1, . . . , uk, vk+1, . . . , vn}) = V,

u_k+1 代 v_i. 前 , t₁, . . . ,t_n∈ F u_k+1=

t1u₁+··· +tku_k+ t_k+1v_k+1+···tnv_n. S^′ linearly independent , tk+1, . . . ,t_n

0. 性, t_k+1̸= 0,

v_k+1= t_k+1⁻¹(uk+1−t1u₁− ··· −tku_k−tk+2v_k+2··· −tnv_n)∈ Span({u1, . . . , uk+1, vk+2, . . . , vn}).

Corollary 1.3.4

Span({u1, . . . , u_k+1, v_k+2, . . . , v_n}) = Span({u1, . . . , u_k, u_k+1, v_k+1, . . . , v_n}) = V.

數 學 , v₁, . . . , v_n ,

Span({u1, . . . , un}) = V. u_n+1 ∈ Span({u1, . . . , un}) S^′ linearly

independent , S^′ linearly dependent. 

Question 1.19. Lemma 1.5.4 S spanning set linearly dependent,

S^′⊆ V, S^′ spanning set 性 ?

Lemma 1.5.4 finite dimensional vector space ,

linearly independent set 數 spanning set 數.

.

Theorem 1.5.5. V finite dimensional vector space. S V basis,

S finite set. S^′ V basis, #(S) = #(S^′).

Proof. V finite dimensional vector space , {v1, . . . , vn} V basis.

Span({v1, . . . , v_n}) = V.

S V basis infinite set, S linearly independent, S

subset linearly independent. u₁, . . . , u_n+1∈ S, {u1, . . . , u_n+1} ⊆ S linearly

independent, Lemma 1.5.4 , S finite set.

S, S^′ V basis, Span(S) = V S^′ linearly independent, Lemma 1.5.4 #(S)≥ #(S^′). Span(S^′) = V S linearly independent,

#(S)≤ #(S^′). #(S) = #(S^′). 

Theorem 1.5.5 finite dimension vector space basis 數

. .

Definition 1.5.6. V finite dimensional vector space over F. S V basis #(S) = n, V over F dimension n, dim(V ) = n.

/0 {O} basis, dim({O}) = 0.

, V over field vector space , dimension

. , dimF(V ) over F dimension. 數 C

overC overR vector space, dim_C(C) = 1 dim_R(C) = 2.

Question 1.20. dim_F(Fⁿ), dim_F(P_n(F)) dim_F(F^S) (S finite set) ? S infinite set, F^S finite dimensional F-space?

Question 1.21. V finite dimensional F-space dim(V ) = n. S^′⊆ V linearly independent S^′′⊆ V V spanning set, #(S^′) #(S^′′) n ?

dimension Proposition 1.5.2 .

Corollary 1.5.7. V finite dimensional vector space over F S^′⊆ V.

.

(1) S V basis.

(2) S V spanning set #(S) = dim(V ).

(3) S linearly independent #(S) = dim(V ).

Proof. dim(V ) = n. S V basis, S V spanning set S linearly independent. dimension #(S) = dim(V ). (1)⇒ (2) (1)⇒ (3).

(2)⇒ (1): S^′( S, #(S^′) < #(S) = n. Span(S^′) = V , Lemma 1.5.4

n linearly independent, dim(V ) = n .

Span(S^′)̸= V Proposition 1.5.2 ((2)⇒ (1)) S V basis.

(3)⇒ (1): S^′′) S, #(S^′′) > #(S) = n. dim(V ) = n V n spanning set, Lemma 1.5.4 S^′′ linearly dependent. Proposition

1.5.2 ((3)⇒ (1)) S V basis. 

V finite dimensional F-space, W V nontrivial F-subspace, W

finite dimensional F-space ? , Lemma 1.5.4

, W basis, Proposition 1.5.2

.

W basis . S1={v1}, v₁∈ W v₁̸= O. S1

linearly independent. Span(S₁) = W , S₁ W basis; Span(S₁)̸=

W , v₂∈ W \ Span(S1). S₂={v1, v₂}, S₂ linearly independent.

Span(S2) = W , S2 W basis; Span(S2)̸= W, 再 v₃∈ W \Span(S2) S₃={v1, v₂, v₃}, Corollary 1.4.4 S₃ linearly independent.

S3, S₄, . . . Si linearly independent. ( n∈ N

Span(S_n) = W ) n∈ N, V n S_n linearly

independent. V finite dimensional vector space, n > dim(V ), Lemma 1.5.4

Sn linearly independent. , m

Span(S_m) = W S_m W basis, W finite dimensional F-space.

.

Theorem 1.5.8. V finite dimensional F-space W V nontrivial F-subspace, W finite dimensional F-space, dim(W ) < dim(V ).

Proof. 前 W finite dimensional F-space. S W basis, S linearly independent, Lemma 1.5.4 dim(W ) = #(S)≤ dim(V). #(S) = dim(V ), Corollary 1.5.7 ((3)⇒ (1)) S V basis, W = Span(S) = V . W

nontrivial subspace , dim(W ) < dim(V ). 

Theorem 1.5.8 , finite dimensional vector space

linearly independent linearly independent,

大 再 大 , Proposition 1.5.2 basis. ,

spanning set spanning set 再

, Proposition 1.5.2 basis. .

Theorem 1.5.9. V finite dimensional F-space S^′⊆ S^′′⊆ V, S^′ linearly independent S^′′ V spanning set, V basis S S^′⊆ S ⊆ S^′′.

Proof. S^′ S^′′\ S^′ linearly independent. V

finite dimensional vector space, Lemma 1.5.4 S^′ 大 .

再 大 S, S^′′\ S Span(S) .

S linearly independent, Span(S) = V , Span(S) = Span(S^′′).

S , S^′′\ S ⊆ Span(S), Corollary 1.3.4 . 

再 Theorem 1.5.9 S , S 數 ,

dim(V ).

Question 1.22. Theorem 1.5.9 finite dimensional vector space linearly independent set 大 basis; spanning set

basis?

vector space , Theorem 1.5.9 (

finite dimensional ) vector space basis.

1.5.2. General Case. vector space basis.

Zorn’s Lemma ,

學, , .

vector space basis, 前 ,

vector space linearly independent, 再 .

( vector space finite dimensional).

Proposition 1.5.2 basis Zorn’s Lemma.

Lemma.

大 (order) ,

大 ( 數 大 , ), totally ordered;

大 ( ), partially ordered.

totally ordered , maximal element ,

大; partially ordered ,

大 . maximal element 大. ,

minimal element . vector space

basis Proposition 1.5.2 vector space spanning set

minimal element; vector space linearly independent set maximal

element. vector space basis , vector space

linearly independent set maximal element ( spanning set

minimal element ). Zorn’s Lemma .

Zorn’s Lemma partially ordered set maximal element . partially ordered set P (ascending chain)

P 大, P maximal element .

前 , Zorn’s Lemma, vector space V linearly

independent set P. P V linearly independent

set. P partial order. P ascending

chain, S1⊆ S2⊆ ··· ⊆ Sn⊆ ··· linearly independent sets ( Si P

V linearly independent set). P S ( S V linearly

independent set) S_i⊆ S, ∀i ∈ N, Zorn’s Lemma P maximal element.

Theorem 1.5.9 .

Theorem 1.5.10. V vector space S^′⊆ S^′′⊆ V, S^′ linearly independent S^′′ V spanning set, V basis S S^′⊆ S ⊆ S^′′.

Proof. P = {S ⊆ V | S is linearly independent,S^′⊆ S ⊆ S^′′}. , S^′∈ P, P nonempty. S₁⊆ S2 ⊆ ···, i∈ N, Si P. T =∪i∈NS_i.

T P , T linearly independent S^′ ⊆ T ⊆ S^′′. Si

S^′⊆ Si⊆ S^′′, S^′ ⊆ T ⊆ S^′′. T linearly independent,

v∈ T v∈ Span(T \ {v}). v₁, . . . , v_n∈ T v_i v v∈ Span({v1, . . . , vn}). v∈ T T =∪i∈NSi v S_k (

S_k+1, S_k+2, . . . ), v_i S_ki . m = max{k,k1, . . . , k_n}, v, v₁, . . . , vn Sm . {v1, . . . , vn} ⊆ Sm\ {v}, v∈ Span(Sm\ {v}), Sm

linearly independent ( S_m P ) T linearly independent.

T P Si⊆ T, ∀i ∈ N, Zorn’s Lemma P maximal

element, S P maximal element.

S V basis, S^′ ⊆ S ⊆ S^′′. S P

, S linearly independent S^′⊆ S ⊆ S^′′, Span(S) = V .

Span(S)̸= V = Span(S^′′), Corollary 1.3.4 w∈ S^′′ w̸∈ Span(S).

S⁺= S∪ {w}. S^′⊆ S⁺⊆ S^′′, Corollary 1.4.4 S⁺ linearly

independent, S⁺ P . S( S⁺, S P maximal

element , Span(S) = V . 

1.6. Direct Sum and Quotient Space

前 subspace , subspace , vector

space vector space . , “ ” vector

space .

1.6.1. Direct Sum. over F vector spaces U,W ( U,W vector space subspaces)

U⊕W = {(u,w) | u ∈ U,w ∈ W}.

the (external) direct sum of U and W . U⊕W , (

). (u1, w1) = (u2, w2), u₁= u2 w₁= w2.

U,W vector space 性 U⊕W F .

(u₁, w₁), (u₂, w₂)∈ U ⊕W r∈ F,

(u₁, w₁) + (u₂, w₂) = (u₁+ u₂, w₁+ w₂) r(u1, w1) = (ru1, rw1)

, U⊕W vector space over F.

Question 1.23. U⊕W vector space over F.

性 ? U⊕W O ( ) ?

Question 1.24. U^′,W^′ U,W subspaces, U^′⊕W^′ U⊕W subspace.

V U⊕W subspace, U,W subspaces U^′,W^′ V = U^′⊕W^′? U,W finite dimensional F-spaces, U⊕W finite dimensional F-space, dimension ?

Proposition 1.6.1. U,W finite dimensional F-spaces, U⊕W finite di- mensional F-space,

dim(U⊕W) = dim(U) + dim(W).

Proof. {u1, . . . , u_m} U basis {w1, . . . , w_n} W basis.

S ={(u1, O_W), . . . , (u_m, O_W), (O_U, w₁), . . . , (O_U, w_n)} U⊕W basis ( O_U, O_W

U,W ).

S U⊕W spanning set. (u, w)∈ U ⊕W, u∈ U

{u1, . . . , um} U basis, c1, . . . , cm∈ F u = c₁u₁+··· + cmu_m. d₁, . . . , d_n∈ F w = d₁w₁+··· + dnw_n.

(u, w) = c₁(u₁, O_W) +··· + cm(u_m, O_W) + d₁(O_U, w₁) +··· + dn(O_U, w_n),

S U⊕W spanning set.

S linearly independent. , S linearly dependent, Proposition 1.4.2 c1, . . . , cm, d1, . . . , dn 0

(O_U, O_W) = c₁(u₁, O_W) +··· + cm(u_m, O_W) + d₁(O_U, w₁) +··· + dn(O_U, w_n),

U⊕W O_U = c₁u₁+··· + cmu_m O_W = d₁w₁+··· + dnw_n. {u1, . . . , u_m} {w1, . . . , w_n} linearly independent c1, . . . , c_m d1, . . . , d_n 0,

前 . S linearly independent. 

over field vector spaces direct sum.

F-spaces direct sum F-spaces direct sum.

Question 1.25. U1, . . . ,U_n F-spaces, U1⊕ ··· ⊕Un ? U1, . . . ,Un finite dimensional F-spaces, dim(U1⊕ ··· ⊕Un) ?

1.6.2. Quotient Space. vector space V subspace W , W V equivalent relation, v₁, v2∈ V, v1∼ v2 v₁− v2∈ W.

equivalent relation.

(1) v∈ V v∼ v: O∈ W, v− v ∈ W.

(2) v₁∼ v2, v₂∼ v1: v₁∼ v2 v₁− v2∈ W, W vector space v₂− v1=−(v1− v2)∈ W, v₂∼ v1.

(3) v₁∼ v2 v₂∼ v3, v₁∼ v3: v₁− v2∈ W v₂− v3∈ W, v₁− v3= (v₁− v2) + (v₂− v3)∈ W.

equivalent relation,

V /W ={v | v ∈ V}.

V /W , u = v u∼ v (

u− v ∈ W).

Question 1.26. 前 ∼ equivalent relation V /W ?

學 代數 group 學 V abelian group, W V

(normal) subgroup, V /W abelian group.

F V /W V /W vector space over F.

u, v∈ V/W r∈ F,

u + v = u + v rv = rv.

W V subspace, V /W well-defined,

V /W vector space over F, the quotient space of V modulo W . Question 1.27. well-defined V /W vector space over F.

V /W ?

Question 1.28. U V subspace W⊆ U, U /W V /W subspace ? W⊆ U

?

V,W finite dimensional F-spaces, V /W finite dimensional F-space, dimension ?

Proposition 1.6.2. V finite dimensional F-spaces W V F-subspace, V /W finite dimensional F-space,

dim(V /W ) = dim(V )− dim(W).

Proof. Theorem 1.5.8 W finite dimensional F-space. {w1, . . . , w_m} W basis, Theorem 1.5.9 v₁, . . . , vn∈ V {w1, . . . , wm, v1, . . . , vn} V basis. S ={v1, . . . , v_n} V /W basis.

S V /W spanning set. v∈ V/W, v∈ V

{w1, . . . , w_m, v₁, . . . , v_n} V basis, c1, . . . , c_m, d₁, . . . , d_n∈ F v = c₁w₁+··· + cmw_m+ d1v₁+··· + dnv_n.

v = c₁w₁+··· + cmw_m+ d1v₁+··· + dnv_n. w_i w_i∈ W, w_i= O,

v = d₁v₁+··· + dnv_n. S V /W spanning set.

S linearly independent. , S linearly dependent, Proposition 1.4.2 d1, . . . , dn∈ F 0 O = d₁v₁+··· + dnv_n.

d₁v₁+··· + dnv_n∈ W = Span({w1, . . . , w_m}),

d1v₁+··· + dnv_n∈ Span({v1, . . . , vn}) ∩ Span({w1, . . . , wm}).

S linearly independent, Corollary 1.4.4

Span({v1, . . . , vn}) ∩ Span({w1, . . . , wm}) = {O},

d₁v₁+···+dnv_n= O. d₁, . . . , d_n 0, {v1, . . . , v_n} linearly independent

. S linearly independent.