大學線性代數初步

大學線性代數初步

大學數學

數學 大 學 線性代數 , 大

數 .

大 學 線性代數學 ,

學 線性代數 . 大學線性代數

. 學 大 大 ( ) 線性

代數 , . 大學 線性代數 , 大

學 數學 . 大 大 數學

, 學 , 線性代數

. , 線性代數

數學 . , .

, .

學 , 數學 學 , 線性

代數 . 線性代數 .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

學 線性代數 學 線性代數 , .

, ,

代. , .

, . , 性

, . , .

v

Chapter 4

Subspaces in R ^m

R^m , 線性 R^m ( 性),

Proposition 1.2.3 8 , vector space. R^m

vector space, R^m subspace. R^m subspaces

性 .

4.1. Subspace and basis

, subspace basis .

R^m V , R^m , Proposition 1.2.3 8

(3), (4) ( O∈ V v∈ V, −v ∈ V) 性 ,

線性 性 ( V 線性 V ), v∈ V,

O =0v∈ V −v = (−1)v ∈ V. V 線性 性

Proposition 1.2.3 8 , R^m vector space

. .

Definition 4.1.1. V R^m . V 線性

V ( v₁, . . . , v_n∈ V c₁, . . . , c_n∈ R c₁v₁+··· + cnv_n∈ V), V

R^m subspace ( ).

/0 subspace. R^m {O} R^m

subspace, R^m R^m subspace.

, Span(v1, . . . , vn) R^m v₁, . . . , vn 線性 ( Definition 1.3.1),

Span(v₁, . . . , v_n) ={v ∈ R^m| r1v₁+··· + rnv_n, for some r₁, . . . , r_n∈ R}.

V R^m subspace v₁, . . . , v_n∈ V

Span(v1, . . . , v_n)⊆ V. (4.1) 69

V subspace, V 線性

V , . , , 線性

.

Proposition 4.1.2. V R^m , V R^m subspace O∈ V

u, v∈ V, r ∈ R u + rv∈ V.

Proof. (⇒) : V , v∈ V. 0v = O, subspace

0v∈ V, O∈ V. subspace , u, v∈ V, r ∈ R

u + rv∈ V.

(⇐) : O∈ V u, v∈ V, r ∈ R u + rv∈ V. O∈ V, V R^m

. (4.1), v₁, . . . , vn∈ V, Span(v1, . . . , vn)⊆ V.

數 n 數學 . ( n = 1),

v₁∈ V, Span(v1) ⊆ V. Span(v1) rv1 ,

r ∈ R rv1∈ V. u = O, v = v₁, rv1 = O + rv1=

u + rv. u + rv∈ V, n = 1 . n = k ,

v₁, . . . , vk ∈ V Span(v1, . . . , vk)⊆ V. v₁, . . . , vk, vk+1∈ V Span(v₁, . . . , v_k, v_k+1)⊆ V. w∈ Span(v1, . . . , v_k+1), c₁, . . . , c_k, c_k+1∈ R

w = c₁v₁+··· + ckv_k+ c_k+1v_k+1. u = c₁v₁+··· + ckv_k, u∈ V.

r = c_k+1∈ R v = v_k+1∈ V, w = u + rv∈ V, Span(v₁, . . . , v_k+1)⊆ V.

數 學 , v₁, . . . , v_n ∈ V Span(v₁, . . . , v_n)⊆ V, V R^m

subspace. 

Proposition 4.1.2, V subspace,

(1) O∈ V

(2) u, v∈ V, r ∈ R ⇒ u + rv ∈ V.

.

Example 4.1.3. a, b, c, d∈ R, S ={(x,y,z) ∈ R³ | ax + by + cz = d}.

a, b, c, d S R³ subspace. S subspace, (x, y, z) = (0, 0, 0) = O∈ S,

d = 0. , d = 0 u = (x, y, z)∈ S, v = (x^′, y^′, z^′)∈ S r∈ R, ax + by + cz = 0, ax^′+ by^′+ cz^′= 0

a(x + rx^′) + b(y + ry^′) + c(z + rz^′) = (ax + by + cz) + r(ax^′+ by^′+ cz^′) = 0,

u + rv = (x + rx^′, y + ry^′, z + rz^′)∈ S. d = 0 d = 0 , S R³ subspace.

Example 4.1.3 , a, b, c, d 0 S =R³, R³ subspace.

R^m subspaces, Proposition 1.3.2 v₁, . . . , vn∈ R^m, Span(v₁, . . . , v_n) R^m subspace ( R^m subspace ).

4.1. Subspace and basis 71

m×n matrix A. A column vectors , A column

space. a₁, . . . , a_n∈ R^m A column vectors, Span(a1, . . . , a_n) A column space.

A row vectors , A row space. A m× n matrix,

column space R^m subspace, row space Rⁿ subspace.

A subspace, homogeneous linear system Ax = O

, {x ∈ Rⁿ| Ax = O}. x = O∈ Rⁿ Ax = O . x = u∈ Rⁿ,

x = v∈ Rⁿ Ax = O , r∈ R,

A(u + rv) = Au + rAv = O,

x = u + rv Ax = O . proposition 4.1.2 {x ∈ Rⁿ| Ax = O} Rⁿ

subspace, A nullspace.

R^m standard basis e₁, . . . , e_m, R^m

e₁, . . . , em 線性 , . standard basis ,

R^m , R^m 性 . R^m

subspace 性 , .

Definition 4.1.4. V R^m subspace. v₁, . . . , v_n∈ V, v∈ V, c₁, . . . , c_n∈ R c₁v₁+··· + cnv_n= v, v₁, . . . , v_n V basis ( ).

Definition 4.1.4 basis ,

basis , .

v₁, . . . , v_n∈ R^m column vectors m× n matrix A, A =



   v ₁ v₂ ··· vn



.

v = c₁v₁+··· + cnv_n x₁= c₁, . . . , x_n= c_n Ax = v , v∈ V, c₁, . . . , c_n∈ R c₁v₁+··· + cnv_n= v, v∈ V

Ax = v . 性 .

Proposition 4.1.5. V R^m subspace v₁, . . . , vn∈ V. A v₁, . . . , vn

column vectors m× n matrix. v₁, . . . , v_n V basis v∈ V,

Ax = v .

, m× n matrix A v∈ V, Ax = v

, A column vectors V basis. A column vectors

V . V basis vectors V .

, V R^m standard basis , standard basis

R^m subspace basis, basis !

v₁, . . . , vn, Proposition 4.1.5 前學

v₁, . . . , v_n V basis.

, basis.

v₁, . . . , v_n ∈ V, V R^m subspace, Span(v₁, . . . , v_n)⊆ V.

v₁, . . . , vn V basis , v∈ V c1, . . . , cn∈ R v =

c₁v₁+··· + cnv_n, v∈ V v∈ Span(v1, . . . , v_n). V ⊆ Span(v1, . . . , v_n),

V = Span(v1, . . . , vn). 性 v₁, . . . , vn, .

Definition 4.1.6. V R^m subspace. v₁, . . . , v_n∈ R^m Span(v1, . . . , v_n) = V , v₁, . . . , v_n V spanning vectors.

v₁, . . . , v_n V spanning vectors, v₁, . . . , v_n∈ V. v₁, . . . , v_n∈ Span(v1, . . . , vn), V = Span(v1, . . . , vn) v₁, . . . , vn∈ V.

v₁, . . . , vn V basis , v∈ V c1, . . . , cn∈ R

c₁v₁+··· + cnv_n= v, V subspace, O∈ V, v = O ,

c1, . . . , c_n∈ R c1v₁+···+cnv_n= O. c1=··· = cn= 0 c1v₁+···+cnv_n= O,

性 , c1, . . . , cn∈ R c1v₁+··· + cnv_n= O. 言 ,

c₁=··· = cn= 0 c₁v₁+··· + cnv_n= O. 性 v₁, . . . , v_n, .

Definition 4.1.7. v₁, . . . , vn∈ R^m. v₁, . . . , vn c1=··· = cn= 0 c₁v₁+··· + cnv_n= O, v₁, . . . , v_n linearly independent.

A v₁, . . . , v_n column vectors m× n matrix, v₁, . . . , v_n linearly independent homogeneous linear system Ax = O nontrivial solution.

x₁= c₁, . . . , x_n= c_n Ax = O nontrivial solution, c₁, . . . , c_n 0 c1v₁+··· + cnv_n= O, v₁, . . . , vn linearly independent .

spanning vectors linear independence 性 , .

, v₁, . . . , v_n∈ V V basis, v₁, . . . , v_n V

spanning vectors linearly independent. basis ,

.

Proposition 4.1.8. V R^m subspace v₁, . . . , vn∈ V. v₁, . . . , vn V basis v₁, . . . , v_n V spanning vectors linearly independent.

Proof. v₁, . . . , v_n V spanning vectors linearly independent

v₁, . . . , v_n V basis. Proposition 4.1.5, A v₁, . . . , v_n

column vectors m× n matrix, v∈ V, Ax = v

. v₁, . . . , v_n V spanning vectors, v₁, . . . , v_n∈ V, A column vectors V , Lemma 3.4.1 V = Span(v1, . . . , vn) v∈ V,

Ax = v . v₁, . . . , v_n linearly independent, homogeneous

4.2. Spanning Vectors and Linear Dependence 73

linear system Ax = O nontrivial solution. Theorem 3.4.6

Ax = v . Proposition 4.1.5 v₁, . . . , v_n V basis. 

Question 4.1. V R^m subspace v₁, . . . , v_n∈ V. A v₁, . . . , v_n column

vectors m× n matrix. v₁, . . . , vn V basis, A column space ? A

nullspace ?

4.2. Spanning Vectors and Linear Dependence

spanning vectors linear independence . ,

步 性 .

4.2.1. Spanning Vectors. R^m subspace V , v₁, . . . , v_n∈ R^m, V spanning vectors v₁, . . . , v_n∈ V. V spanning vectors,

V .

, 數 , spanning vectors . Corollary

3.4.4 , n < m ,R^m n R^m spanning vectors.

v₁, . . . , vn 數 , v_n+1 subspace Span(v1, . . . , vn, vn+1) subspace Span(v₁, . . . , v_n), 大,

subspace 大.

Lemma 4.2.1. v₁, . . . , v_n, v_n+1∈ R^m. Span(v₁, . . . , v_n)⊆ Span(v1, . . . , v_n, v_n+1), Span(v1, . . . , vn)̸= Span(v1, . . . , vn, vn+1) v_n+1̸∈ Span(v1, . . . , vn).

Proof. Span(v₁, . . . , v_n, v_n+1) R^m subspace, v₁, . . . , v_n∈ Span(v1, . . . , v_n, v_n+1), (4.1)

Span(v₁, . . . , v_n)⊆ Span(v1, . . . , v_n, v_n+1). (4.2) v_n+1∈ Span(v1, . . . , v_n), Span(v₁, . . . , v_n) R^m subspace, v₁, . . . , v_n, v_n+1∈ Span(v1, . . . , v_n)

Span(v₁, . . . , v_n, v_n+1)⊆ Span(v1, . . . , v_n). (4.3) (4.2), (4.3)

Span(v1, . . . , vn) = Span(v1, . . . , vn, vn+1).

Span(v₁, . . . , v_n)̸= Span(v1, . . . , v_n, v_n+1) v_n+1̸∈ Span(v1, . . . , v_n).

, v_n+1̸∈ Span(v1, . . . , v_n), v_n+1∈ Span(v1, . . . , v_n, v_n+1), Span(v₁, . . . , v_n)̸=

Span(v1, . . . , vn, vn+1). 

V Rⁿ subspace. v₁, . . . , v_n ∈ V V spanning vectors , Span(v₁, . . . , v_n)̸= V, V v_n+1 v_n+1̸∈ Span(v1, . . . , v_n). Lemma 4.2.1 , Span(v1, . . . , vn, vn+1) Span(v1, . . . , vn) 大,

, V spanning vectors.

4.2.2. Linear Dependence. linearly independent ,

linearly dependent linear independent , “線

性 ”.

linearly dependent, ,

線性 . v₁, . . . , v_n linearly dependent, v_i

v₁, . . . , vi−1, vi+1, . . . , vn 線性 . v_i 線性

. 步 , v_i= r₁v₁+··· + rn−1v_n−1+ r_n+1v_n+1+··· + rnv_n,

rj 數.

r1v₁+··· + rn−1v_n−1− vi+ rn+1v_n+1+··· + rnv_n= O.

0 數 c₁, . . . , c_n c₁v₁+···+cnv_n= O. , 0 數 c₁, . . . , cn c1v₁+··· + cnv_n= O. ci̸= 0,

v_i=−c1

ci

v₁+··· +−ci−1

ci

v_i−1+−ci+1

ci

v_i+1+··· +−cn

ci

v_n,

v_i v₁, . . . , v_i−1, v_i+1, . . . , v_n 線性 . , 0

數 c₁, . . . , c_n c₁v₁+···+cnv_n= O v₁, . . . , v_n .

線性 線性 , .

線性 .

Definition 4.2.2. v₁, . . . , v_n∈ R^m, 0 數 c₁, . . . , c_n c1v₁+··· + cnv_n= O,

v₁, . . . , v_n linearly dependent.

v₁, . . . , vn∈ R^m Definition 4.2.2, 0 數 c₁, . . . , cn

c₁v₁+··· + cnv_n= O, v₁, . . . , v_n 線性 .

Definition 4.1.7 linearly independent . linearly

independent linear dependent . A v₁, . . . , v_n column vectors m×n matrix. v₁, . . . , v_n linearly independent homogeneous linear system Ax = O nontrivial solution, v₁, . . . , vn linearly dependent

homogeneous linear system Ax = O nontrivial solution.

v₁, . . . , v_n linearly independent, :

c1v₁+··· + cnv_n= O, c1, . . . , cn 0. ,

, v₁, . . . , v_n linearly dependent ( 0

數 c₁, . . . , c_n c₁v₁+··· + cnv_n= O), .

, 大 , .

Example 4.2.3. v₁, . . . , v_n∈ R^m linearly independent. v₁, . . . , v_n

線性 . v₁, . . . , v_n v_n, v₁, . . . , v_n−1

linearly independent. , , c1v₁+··· +

c_n−1v_n−1= O c₁, . . . , c_n−1 0. , 0

4.2. Spanning Vectors and Linear Dependence 75

數 c₁, . . . , c_n−1 c₁v₁+··· + cn−1v_n−1= O. c_n= 0, 0 數

c1, . . . , c_n

c₁v₁+··· + cn−1v_n−1+ c_nv_n= c₁v₁+··· + cn−1v_n−1= O.

v₁, . . . , v_n∈ R^m linearly independent , v₁, . . . , v_n−1 linearly independent. 大 , v₁, . . . , vn−1∈ R^m linearly dependent

, v_n∈ Rⁿ , v₁, . . . , v_n−1, v_n linearly dependent.

Example 4.2.3 , v₁, . . . , v_n∈ R^m linearly independent

, , linearly independent 性 .

. linearly

independent.

Lemma 4.2.4. v₁, . . . , v_n, v_n+1∈ R^m. v₁, . . . , v_n linearly independent, v₁, . . . , v_n, v_n+1 linearly independent v_n+1̸∈ Span(v1, . . . , v_n).

Proof. v_n+1∈ Span(v1, . . . , v_n), v₁, . . . , v_n, v_n+1 線性 , linearly de- pendent. v₁, . . . , vn, vn+1 linearly independent, v_n+1∈ Span(v1, . . . , vn)

. v_n+1̸∈ Span(v1, . . . , v_n).

, v_n+1̸∈ Span(v1, . . . , vn) v₁, . . . , vn, vn+1 linearly independent.

, v₁, . . . , v_n, v_n+1 linearly dependent, 0 數

c1, . . . , cn, cn+1 c1v₁+··· + cnv_n+ cn+1v_n+1= O. cn+1 0, v_n+1= −c1

cn+1

v₁+··· + −cn

cn+1

v_n,

v_n+1∈ Span(v1, . . . , v_n) . c_n+1= 0, c₁, . . . , c_n 0 數

c₁v₁+··· + cnv_n= c₁v₁+··· + cnv_n+ c_n+1v_n+1= O,

v₁, . . . , v_n linearly dependent. v₁, . . . , v_n linearly independent

, v₁, . . . , vn, vn+1 linearly independent. 

V Rⁿ subspace, v₁, . . . , vn∈ V linearly independent. v₁, . . . , vn

V spanning vectors, V v_n+1 v_n+1̸∈ Span(v1, . . . , v_n).

Lemma 4.2.1 v₁, . . . , v_n, v_n+1 linearly independent.

, 數 , linearly independent . Corollary

3.4.8 , n > m ,R^m n linearly dependent.

.

Lemma 4.2.5. w₁, . . . , wk∈ R^m. v₁, . . . , vn∈ Span(w1, . . . , wk) n > k, v₁, . . . , vn

linearly dependent.

Proof. v₁, . . . , v_n∈ Span(w1, . . . , w_k) a_{i j} 1≤ i ≤ n, 1 ≤ j ≤ k v₁ = a_{1 1}w₁ + a_{1 2}w₂ + ··· + a1 kw_k

...

v_n = a_{n 1}w₁ + a_{n 2}w₂ + ··· + an kw_k. c1, . . . , cn∈ R

c₁v₁ = c₁a_{1 1}w₁ + c₁a_{1 2}w₂ + ··· + c1a_{1 k}w_k ...

c_nv_n = c_na_{n 1}w₁ + c_na_{n 2}w₂ + ··· + cna_{n k}w_k. , n× k matrix A = [ai j] c₁v₁+··· + cnv_n= b₁w₁+··· + bkw_k

[ c1 ··· cn

]A =[

b1 ··· bk

], A^T



 c₁

... c_n



 =



 b₁

... b_k



.

A^T k×n matrix n > k, Corollary 3.4.7 homogeneous linear system A^Tx = O nontrivial solution. c1, . . . , cn∈ R 0, A^T



 c₁

... c_n



 =



 0

... 0



, [ c1 ··· cn

]A =[

0 ··· 0 ]

. 0 c1, . . . , ck

c1v₁+··· + cnv_n= 0w1+··· + 0wk= O.

v₁, . . . , v_n linearly dependent. 

Question 4.2. v₁, . . . , v_n linearly independent. w₁, . . . , w_k∈ Span(v1, . . . , v_n) k < n, Span(w₁, . . . , w_k)̸= Span(v1, . . . , v_n).

4.3. Dimension of subspace

R^m subspace basis, .

R^m nonzero subspace basis. subspace

basis 數 , 數 subspace dimension.

dimension 性 .

V R^m subspace. V ={O}, O V , O V

spanning vector. O linearly independent, c̸= 0

cO = O. V basis . V ̸= {O} , v₁∈ V v₁̸= O.

V₁= Span(v₁). V₁= V , v₁ V spanning vector, v₁ linearly independent, v₁ V basis. V1̸= V, v₂∈ V v₂̸∈ V1= Span(v₁).

V₂= Span(v₁, v₂). Lemma 4.2.1, V₁⊆ V2 V₁̸= V2, V₂ V₁ 大. Lemma 4.2.4 v₁, v₂ linearly independent. V₂= V ,

v₁, v2 V spanning vectors, linearly independent, v₁, v2

V basis. V₂̸= V, 步 , .

4.3. Dimension of subspace 77

Theorem 4.3.1. V R^m subspace V̸= {O}, v₁, . . . , v_n∈ V, n≤ m,

V basis.

Proof. 前 , V₂̸= V , . , m 大,

前 步 步 , 數學 . 前

, i vectors v₁, . . . , v_i linearly independent, 步

, n n≤ m Span(v1, . . . , v_n) = V .

數學 , k v₁, . . . , v_k linearly independent (

k = 1 ). Vk= Span(v1, . . . , vk). Vk= V , v₁, . . . , vk V spanning vectors, linearly independent, v₁, . . . , v_k V basis. V_k ̸= V, Vk ⊆ V v_k+1∈ V v_k+1̸∈ Vk. v₁, . . . , v_k, v_k+1, v₁, . . . , v_k linearly independent v_K+1̸∈ Vk Lemma 4.2.4 v₁, . . . , v_k, v_k+1 linearly independent.

數學 步 vectors linearly independent.

, 步 n≤ m, Span(v₁, . . . , v_n) = V .

m , Span(v1, . . . , vm) = V , 前 數學 v₁, . . . , vm linearly independent. v_m+1 ∈ V v_m+1̸∈ Span(v1, . . . , v_m), Lemma 4.2.4 v₁, . . . , v_m, v_m+1 linearly independent. v₁, . . . , v_m+1∈ V v₁, . . . , v_m+1∈ R^m,

Corollary 3.4.8 v₁, . . . , vm+1 m + 1 linearly dependent.

v₁, . . . , v_m, v_m+1 linearly independent . m

Span(v1, . . . , vm) = V , n≤ m v₁, . . . , vn V spanning vectors. v₁, . . . , v_n linearly independent, v₁, . . . , v_n V

basis. 

Theorem 4.3.1 , v₁. . . v_k∈ V linearly independent,

v₁, . . . , vk 大 V basis. v₁, . . . , vk V spanning vectors, 前 , v_k+1, . . . , v_n ∈ V v₁, . . . , v_k, v_k+1, . . . , v_n V

basis.

Theorem 4.3.1 basis 性, basis

性. R^m subspace basis .

[ 1 0

] ,

[ 0 1

] [

1 1

] ,

[ −1 1

]

R² basis.

v₁, . . . , vn R^m basis, Corollary 3.4.4 n < m, v₁, . . . , vn

R^m spanning vectors, n≥ m. n > m, Corollary 3.4.8 v₁, . . . , v_n

linearly independent, n = m. R^m basis m

. .

Theorem 4.3.2. V R^m subspace. v₁, . . . , vn w₁, . . . , wk V basis, n = k.

Proof. , n̸= k. 性, n > k. w₁, . . . , w_k V spanning vectors, v₁, . . . , v_n∈ V = Span(w1, . . . , w_k). n > k Lemma 4.2.5

v₁, . . . , vn linearly dependent. v₁, . . . , vn V basis .

n̸= k, n = k. 

Theorem 4.3.2 V basis 數 . n

V basis, V basis n .

.

Definition 4.3.3. V R^m subspace. V basis 數 V

dimension ( ), dim(V ) .

R^m basis m , dim(R^m) = m.

Theorem 4.3.1 , V linearly independent

linearly independent spanning vectors

. , V spanning vectors, spanning vectors

linearly independent . ,

.

Proposition 4.3.4. V R^m subspace v₁, . . . , v_n∈ V.

(1) v₁, . . . , v_n V spanning vectors, dim(V )≤ n. , v₁, . . . , v_n linearly dependent, dim(V ) < n.

(2) v₁, . . . , v_n linearly independent, dim(V )≥ n. , v₁, . . . , v_n V spanning vectors, dim(V ) > n.

Proof. dim(V ) = k, w₁, . . . , w_k V basis.

(1) v₁, . . . , vn V spanning vectors, Span(v1, . . . , vn) = V . w₁, . . . , w_k V basis, w₁, . . . , w_k∈ Span(v1, . . . , v_n) w₁, . . . , w_k linearly independent.

k > n, Lemma 4.2.5 w₁, . . . , wk linearly dependent. ,

dim(V ) = k≤ n. v₁, . . . , v_n linearly dependent, v₁, . . . , v_n

線性 . 性 v_n, v_n∈ Span(v1, . . . , v_n−1).

Lemma 4.2.1 Span(v1, . . . , vn−1) = Span(v1, . . . , vn−1, vn) = V , v₁, . . . , vn−1 V spanning vectors. dim(V )≤ n − 1, dim(V ) < n.

(2) v₁, . . . , v_n∈ V linearly independent. w₁, . . . , w_k V basis, Span(w1, . . . , w_k) = V , v₁, . . . , v_n∈ Span(w1, . . . , w_k). n > k, Lemma 4.2.5

v₁, . . . , v_n∈ V linearly dependent . n≤ k = dim(V). v₁, . . . , v_n V spanning vectors, v_n+1∈ V v_n+1̸∈ Span(v1, . . . , v_n) Lemma 4.2.4

v₁, . . . , vn, vn+1 linearly independent. dim(V )≥ n + 1,

dim(V ) > n. 

4.3. Dimension of subspace 79

v₁, . . . , v_n V basis, v₁, . . . , v_n V

spanning vectors linearly independent . dim(V )

n, spanning vectors linearly independent

.

Corollary 4.3.5. V R^m subspace v₁, . . . , vn∈ V. . (1) v1, . . . , vn V basis.

(2) dim(V ) = n v₁, . . . , v_n V spanning vectors.

(3) dim(V ) = n v₁, . . . , v_n linearly independent.

Proof. (1)⇒ (2): v₁, . . . , v_n V basis, V basis n , dim(V ) = n. basis spanning vectors, v₁, . . . , v_n V spanning vectors. (1)⇒ (2).

(2)⇒ (3): v₁, . . . , vn V spanning vectors, Proposition 4.3.4 (1) dim(V )≤ n. v₁, . . . , v_n linearly independent, Proposition 4.3.4 (1)

dim(V ) < n. dim(V ) = n , v₁, . . . , vn linearly independent.

(2)⇒ (3).

(3)⇒ (1): v₁, . . . , vn linearly independent, Proposition 4.3.4 (2) dim(V )≥ n. v₁, . . . , v_n V spanning vectors, Proposition 4.3.4 (2) dim(V ) >

n. dim(V ) = n , v₁, . . . , vn V spanning vectors. v₁, . . . , vn V

basis, (3)⇒ (1). 

Theorem 4.3.1 V R^m subspace, V basis 數

m, dim(V )≤ m, dim(V )≤ dim(R^m).

.

Corollary 4.3.6. V,W R^m subspaces V ⊆ W, dim(V )≤ dim(W).

dim(V ) = dim(W ) V = W .

Proof. dim(V ) = n v₁, . . . , v_n V basis. v₁, . . . , v_n∈ V V ⊆ W, v₁, . . . , v_n∈ W. v₁, . . . , v_n∈ W linearly independent Proposition 4.3.4 (2) dim(V ) = n≤ dim(W).

V = W , dimension 性 dim(V ) = dim(W ). , dim(V ) = dim(W )

V ̸= W. v₁, . . . , v_n∈ W linearly independent W spanning vectors, Proposition 4.3.4 (2) dim(V ) = n < dim(W ). dim(V ) = dim(W ) ,

V = W . 

dim(V ) = dim(W ) 代 V = W . R² v̸= O, Span(v)

dimension 1 subspace. w̸= O v , Span(v)̸= Span(w),

dimension 1 subspace. Corollary 4.3.6 V ⊆ W 前 dim(V ) = dim(W ) V = W .

4.4. Column Space and Nullspace

column space, row space nullspace basis.

column space row space dimension rank.

subspace basis.

, column space nullspace 數

. column space nullspace 性, :

Definition 4.4.1. A =



   a ₁ a₂ ··· an



 R^m a₁, . . . , a_n column

vectors m× n matrix.

(1) Span(a1, . . . , an) A column space, C(A) A column space.

(2) homogeneous linear system Ax = O A nullspace

N(A) A nullspace. N(A) ={x ∈ Rⁿ| Ax = O}.

Lemma 3.4.1 Theorem 3.4.6 .

Proposition 4.4.2. A m× n matrix b∈ R^m, Ax = b.

(1) Ax = b b∈ C(A).

(2) Ax = b N(A) ={O}.

column space nullspace sub-

spaces basis. R^m subspace V basis, V

spanning vectors. spanning vectors linearly independent

. , lin-

early independent. , linearly

independent, .

Example 4.4.3.

v₁=





 0 3 0 1





,v²=





 2

−5 0 0





,v³=





 0 7

−1 0





.

4.4. Column Space and Nullspace 81

v₁, v₂, v₃ linearly independent, c₁= c₂= c₃= 0 , c1v₁+ c₂v₂+ c₃v₃= O.

c1v₁+ c2v₂+ c3v₃= c1





 0 3 0 1





 + c²





 2

−5 0 0





 + c³





 0 7

−1 0





 =







2c2

3c1− 5c2+ 7c3

−c3

c₁





.

c₁v₁+ c₂v₂+ c₃v₃= O, c₁v₁+ c₂v₂+ c₃v₃ 1-st entry 2c₂, 3-rd entry

−c3 4-th entry c1 0, c1= c2= c3= 0. c1= c2= c3= 0 , c₁v₁+ c₂v₂+ c₃v₃= O, v₁, v₂, v₃ linearly independent.

Example 4.4.3 , v₁, . . . , vn v_i entry

0, entry 0, v₁, . . . , v_n linearly independent. ( Example 4.4.3 v₁ 4-th entry 1, v₂, v3 4-th entry 0; v2 1-st entry 2, v₁, v3

1-st entry 0; v₃ 3-rd entry −1, v₁, v₂ 3-th entry 0, ).

v_i 0 entry ai, c1v₁+··· + cnv_n entry ciai, c1v₁+··· + cnv_n= O, ciai = 0, ci 0. v₁, . . . , vn linearly independent.

A m×n matrix, A nullspace N(A) homogeneous linear system Ax = O

. Ax = O , nullspace

basis .

Ax = O , elementary row operations A

echelon form ( reduced echelon form) A^′. A^′x = O Ax = O ,

A A^′ nullspace. free variable, free variable

代 數, . , pivot variable free

variables , free variable , . free

variables xi1, . . . , xi_k. j = 1, . . . , k, xij= 1, free variable 0 , v_j. v_j i_j-th entry 1, v_ij′ i_j-th entry 0, 前 v₁, . . . , vk linearly independent. r1, . . . , rk∈ R, r1v₁+··· + rkv_k

free variables x_i1, . . . , x_ik 代 x_i1 = r₁, . . . , x_ik= r_k . 言

r1v₁+··· + rkv_k , v₁, . . . , v_k A nullspace spanning vectors. v₁, . . . , v_k A nullspace basis, A nullspace

dimension free variables 數, A column 數 pivot 數,

.

Proposition 4.4.4. A m× n matrix. row operations A echelon

form A^′ , A^′ pivot 數 r, A nullspace dimension n− r. A^′x = O

free variables xi1, . . . , xik. j = 1, . . . , k, xij= 1, free variable 0, v_j. v₁, . . . , v_k A nullspace basis.

nullspace echelon form , nullspace

dimension , Proposition 4.4.4 “

elementary row operations echelon form , pivot 數

”. Definition 2.3.1 rank well-defined.

Example 4.4.5. A nullspace,

A =







2 1 1 0 0 0

1 0 0 1 0 0

1 1 1 0 1 2

1 2 2 −2 1 2





.

A 2-nd row −2, −1, −1 1-st, 3-rd 4-th row, 1-st, 2-nd

rows 





1 0 0 1 0 0

0 1 1 −2 0 0

0 1 1 −1 1 2

0 2 2 −3 1 2





.

2-nd row −1,−2 3-rd 4-th row







1 0 0 1 0 0

0 1 1 −2 0 0

0 0 0 1 1 2

0 0 0 1 1 2





.

3-rd row −1 4-th row, echelon form







1 0 0 1 0 0

0 1 1 −2 0 0

0 0 0 1 1 2

0 0 0 0 0 0





.

homogeneous linear system

x1 +x4 = 0

x₂ +x₃ −2x4 = 0

+x₄ +x₅ +2x₆ = 0

. echelon form x₁, x₂, x₄ pivot variable, x₃, x₅, x₆ free variable.

x6= 1, x5= 0, x3= 0, x4=−2, x2=−4, x1= 2, x6= 0, x5= 1, x3= 0 x4=−1, x₂=−2, x1= 1, x₆= 0, x₅= 0, x₃= 1 x₄= 0, x₂=−1, x1= 0.

v₁=









 2

−4 0

−2 0 1









 , v2=









 1

−2 0

−1 1 0









 , v3=









 0

−1 1 0 0 0











4.4. Column Space and Nullspace 83

A nullspace basis. , x₆, x₅, x₃ 數 r, s,t,

x4=−2r − s, x2=−4r − 2s −t, x1= 2r + s. A nullspace











2r + s

−4r − 2s −t t

−2r − s s r











= r









 2

−4 0

−2 0 1









 + s









 1

−2 0

−1 1 0









 + t









 0

−1 1 0 0 0











= rv₁+ sv₂+ tv₃.

v₁, v₂, v₃ A nullspace spanning vectors, v₁, v₂, v₃ linearly independent, v₁, v2, v3 N(A) basis.

Question 4.3. Example 4.4.5 A reduced echelon form.

N(A) basis ?

matrix A column space C(A) basis.

A column space, Ax = v v .

v, A column space. A 數

constrain equations v. ,

, .

Example 4.4.6. Example 4.4.5 4×6 matrix A. A column vectors

basis. b A column space , Ax = b ,

b =





 b₁ b2

b3

b4





,

b₁, b₂, b₃, b₄ .

2x1 +x2 +x3 = b1

x₁ +x₄ = b₂

x₁ +x₂ +x₃ +x₅ +2x₆ = b₃

x₁ +2x₂ +2x₃ −2x4 +x₅ +2x₆ = b₄

augmented matrix [A| b], Example 4.4.5 elementary row operations







2 1 1 0 0 0 b1

1 0 0 1 0 0 b2

1 1 1 0 1 2 b3

1 2 2 −2 1 2 b4





 ∼







1 0 0 1 0 0 b2

0 1 1 −2 0 0 b1− 2b2

0 1 1 −1 1 2 b3− b2

0 2 2 −3 1 2 b4− b2





 ∼







1 0 0 1 0 0 b₂

0 1 1 −2 0 0 b1− 2b2

0 0 0 1 1 2 b3+ b2− b1

0 0 0 1 1 2 b4+ 3b2− 2b1





 ∼







1 0 0 1 0 0 b₂

0 1 1 −2 0 0 b1− 2b2

0 0 0 1 1 2 b3+ b2− b1

0 0 0 0 0 0 b4− b3+ 2b2− b1





.

( 2.1 (2)(a)(b) ) , Ax = b

b₄− b3+ 2b₂− b1= 0. 言 , b₁− 2b2+ b₃− b4= 0 , b