大學線性代數再探

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 (

).

, ,

代. , .

, . , 性

, . , .

, 大 . 大

, . ,

.

v

Chapter 1

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)

?

1.1. Definition and Basic Properties 3

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, .

1.2. Subspaces 5

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 7

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 9

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.

1.5. Basis and Dimension 11

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. Basis and Dimension 13

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 ).

1.5. Basis and Dimension 15

(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 .

1.6. Direct Sum and Quotient Space 17

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),

1.6. Direct Sum and Quotient Space 19

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.