Linear Transformation of General Vector

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Linear Transformation of General Vector

Space

, 前 vector space. vector space

vector space linear transformation. 前 , vector space

性 , Rⁿ . 前 .

6.1. Vector Space and Subspace

R^m , 線性 R^m ,

Proposition 1.2.3 8 , vector space.

vector space vector space subspaces 性 .

V , V +, V u, v∈ V,

u + v V ( 性). R V

數 r∈ R v∈ V, r v rv V ( 數

性).

Definition 6.1.1. V +, R V 數 .

8 性 , V vector space.

(1) u, v∈ V, u + v = v + u.

(2) u, v, w∈ V, (u + v) + w = u + (v + w).

(3) O∈ V u∈ V O + u = u.

(4) u∈ V u^′∈ V u + u^′= O.

(5) r, s∈ R u∈ V, r(su) = (rs)u.

(6) r, s∈ R u∈ V, (r + s)u = ru + su.

111

(7) r∈ R u, v∈ V r(u + v) = ru + rv.

(8) u∈ V, 1u = u.

, vector space,

數 . vector space , V vector

space + , 數 數 ,

數 . vector space, Rⁿ,

數 , 數 .

vector space , 數 . vector space

數 數 , “field” 數 . over

field vector space ( over field 大). 學

field 數 R ,

overR vector space, vector space .

Corollary 1.2.4 Rⁿ , 性 (3) O (4)

u, u^′ . Rⁿ 數 ,

8 性 , vector space .

Proposition 6.1.2. V vector space, V O

u∈ V O + u = u. , u∈ V, u^′∈ V u + u^′= O.

O , O V O + u = u,∀u ∈ V

, V additive identity zero element. u∈ V,

u^′ u + u^′= O, −u u^′,

u additive inverse.

, 性 (3) O u∈ V

O + u = u ( Proposition 6.1.2 u ),

u∈ V w + u = u, w = O. Proposition 6.1.2

, u∈ V, w + u = u, w = O . 性

O 性 . 數 r, rO = O.

性 (7) rO + rO = r(O + O) = rO. rO rO rO, 前

rO = O. , , Corollary 1.2.6

Corollary 1.2.7.

Proposition 6.1.3. V vector space, .

(1) w∈ V u∈ V w + u = u, w = O.

(2) v∈ V 0v = O.

(3) r∈ R rO = O.

(4) r∈ R,v ∈ V (−1)(rv) = −(rv) = r(−v).

Proposition 6.1.3 r = 0 v = O rv = O, r̸= 0 v̸= O,

rv = O ? . r̸= 0 v̸= O, 1/r rv

vector space 性 (5) ¹_r(rv) =^r_rv =1v, 性 (8) 1v = v, ¹_r(rv) = v̸= O.

rv = O, Proposition 6.1.3 (3) ¹_r(rv) = ¹_rO = O, . rv O.

Question 6.1. V vector space, v∈ V v̸= O, r, s∈ R r̸= s,

rv̸= sv, V vector space over R V , V

.

言 , vector space 數 8 性 , vector

space 數 . , 數 “ ” ,

w + (−v) w− v. 大

. 2u + v = w, 1/2 u = ¹₂(w− v).

vector space .

Example 6.1.4. (A) Mm×n m× n matrices .

數 ( Definition 3.1.2), vector space

8 性 ( Proposition 3.1.3). 數 Mm×n

vector space.

(B) P(R) x 數 數 . 數 ,

P(R) vector space. f (x) = anxⁿ+··· + a1x + a0, g(x) = b_mx^m+··· + b1x + b0∈ P(R), m≤ n, g(x) g(x) = bnxⁿ+···+bmx^m+···+b1x + b0,

b_n = b_n−1 =··· = bm+1= 0. 數 ,

數 . f (x) + g(x)

f (x) + g(x) =∑ⁿi=0(a_i+ b_i)xⁱ. r∈ R, 數 r f (x) r f (x) =∑ⁿi=0(ra_i)xⁱ.

, 數 數 ,

數 P(R) . vectors

space 8 .

(1) f (x) =∑ⁿi=0aixⁱ, g(x) =∑ⁿi=0bixⁱ∈ P(R) f (x) + g(x) =

∑

(a_i+ b_i)xⁱ=

∑

(b_i+ a_i)xⁱ= g(x) + f (x).

(2) f (x) =∑ⁿi=0aixⁱ, g(x) =∑ⁿi=0bixⁱ, h(x) =∑ⁿi=0cixⁱ∈ P(R) ( f (x) + g(x)) + h(x) =

∑

(a_i+ b_i)xⁱ+

∑

c_ixⁱ=

∑

((a_i+ b_i) + c_i)xⁱ,

f (x) + (g(x) + h(x)) =

∑

a_ixⁱ+

∑

(b_i+ c_i)xⁱ=

∑

(a_i+ (b_i+ c_i))xⁱ.

(a_i+ b_i) + c_i= a_i+ (b_i+ c_i), ( f (x) + g(x)) + h(x) = f (x) + (g(x) + h(x)).

(3) g(x) = 0 =∑ⁿi=0b_ixⁱ∈ P(R), b_i= 0, ∀i = 0,1,...,n.

f (x) =∑ⁿi=0aixⁱ∈ P(R), f (x) + g(x) =

∑

(ai+ bi)xⁱ=

∑

aixⁱ= f (x).

(4) f (x) =∑ⁿi=0a_ixⁱ∈ P(R), h(x) =∑ⁿi=0(−ai)xⁱ∈ P(R), f (x) + h(x) =

∑

(a_i− ai)xⁱ=

∑

0xⁱ= 0.

(5) r, s∈ R f (x) =∑ⁿi=0a_ixⁱ∈ P(R), r(s f (x)) = r(

∑

(sa_i)xⁱ) =

∑

(r(sa_i))xⁱ=

∑

((rs)a_i)xⁱ= (rs)

∑

a_ixⁱ= (rs) f (x).

(6) r, s∈ R f (x) =∑ⁿi=0a_ixⁱ∈ P(R), (r + s) f (x) =

∑

((r + s)a_i)xⁱ=

∑

(ra_i+ sa_i)xⁱ=

∑

(ra_i)xⁱ+

∑

(sa_i)xⁱ= r f (x) + s f (x).

(7) r∈ R f (x) =∑ⁿi=0aixⁱ, g(x) =∑ⁿi=0bixⁱ∈ P(R) r( f (x) + g(x)) = r(

∑

(a_i+ b_i)xⁱ) =

∑

(r(a_i+ b_i))xⁱ=

∑

(ra_i+ rb_i)xⁱ= r f (x) + rg(x).

(8) f (x) =∑ⁿi=0aixⁱ∈ P(R), 1 f (x) =

∑

(1ai)xⁱ=

∑

aixⁱ= f (x).

P(R) 數 vector space 8 , 數

P(R) vector space.

(C) L (Rⁿ,R^m) Rⁿ R^m linear transformations .

L (Rⁿ,R^m) 數 , 數 L (Rⁿ,R^m) (

Proposition 5.1.6). linear transformation 數

代 , R^m vector space, L (Rⁿ,R^m)

數 vector space 8 . (2), T1, T₂, T₃∈ L (Rⁿ,R^m).

v∈ Rⁿ,

((T₁+ T₂) + T₃)(v) = (T₁+ T₂)(v) + T₃(v) = (T₁(v) + T₂(v)) + T₃(v),

(T1+ (T2+ T3))(v) = T1(v) + (T2+ T3)(v) = T1(v) + (T2(v) + T3(v)).

R^m , ((T₁+ T₂) + T₃)(v) = (T₁+ (T₂+ T₃))(v).

v∈ Rⁿ , 數 (T₁+ T₂) + T₃= T₁+ (T₂+ T₃).

(3) zero element zero mapping, T :Rⁿ→ R^m, T (v) = O,

∀v ∈ Rⁿ linear transformation. L (Rⁿ,R^m) 數 vector space

8 , 數 L (Rⁿ,R^m) vector space.

Rⁿ , Rⁿ Rⁿ 數 vector

space, Rⁿ subspace. vector space ,

vector space 數 vector space, vector space

subspace.

Definition 6.1.5. V vector space, W V nonempty subset. V

數 W vector space, W V subspace.

vector space subspace vector space, subspace

vector space 8 . vector space 8

(3)(4) , ,

. Rⁿ subspace 性 ( Proposition 4.1.2).

Proposition 6.1.6. V vector space W V . W V subspace O∈ W u, v∈ W, r ∈ R u + rv∈ W.

Proof. (⇒) : subspace , 數 性, u, v∈ W, r ∈ R

u + rv∈ W. W , w∈ W. 0w, 性 0w∈ W.

V vector space, 0w = O (Proposition 6.1.3(2)). O =0w∈ W.

(⇐) : O∈ W, W V . u, v∈ W, r ∈ R

u + rv∈ W, V 數 W 性, W

數 . W V subspace, 數 W

vector space 8 . (3)(4) , V

. W V , W V , W

. O∈ W, (3) . w∈ W, 數 性 (−1)w ∈ W,

w + (−1)w = 0w = O, (4) . 

Proposition 6.1.6, vector space V W V

subspace, (1) O∈ V

(2) u, v∈ V, r ∈ R ⇒ u + rv ∈ V.

. .

Example 6.1.7. (A) Mn×n, n× n vector space.

Mn×n symmetric matrices ( ) Mn×n subspace.

A∈ Mn×n symmetric matrix, A^T= A. n× n O

symmetric matrix. A, B∈ Mn×n A^T = A, B^T = B, r ∈ R,

(A + rB)^T = A^T+ (rB)^T= A + rB ( Proposition 3.2.4). A + rB symmetric

matrix, Mn×n symmetric matrices Mn×n subspace.

Mn×n invertible matrices ( ) Mn×n subspace

? . O invertible, O

Mn×n invertible matrices Mn×n subspace.

invertible matrices {O} , Mn×n subspace.

O , invertible matrices invertible. 2× 2

,

[ 1 0 0 1

] [

0 1 1 0

]

invertible,

[ 1 0 0 1

] +

[ 0 1 1 0

]

=

[ 1 1 1 1

]

invertible.

(B) P(R), x 數 數 vector space.

數 n∈ N, 數 n Pn(R) P(R) subspace.

P_n(R) P_n(R) = {∑ⁿi=0a_ixⁱ| ai∈ R}. P_n(R)

( 數學 數 −∞, 0. 代數

, ). f (x) =∑ⁿi=0a_ixⁱ, g(x) =∑ⁿi=0b_ixⁱ∈ Pn(R), r∈ R, f (x) + rg(x) =∑ⁿi=0(a_i+ rb_i)xⁱ∈ Pn(R). Pn(R) P(R) subspace. ,

數 n , P(R) subspace .

. , 數 n 數

, (x²+ x + 1) + (−x²+ x + 1) = 2x + 2. ,

vector space.

(C) L (Rⁿ,R^m) Rⁿ R^m linear transformations vector space.

Rⁿ subspace V ,

N(V ) ={T ∈ L (Rⁿ,R^m)| V ⊆ ker(T)}.

N(V ) v∈ V, T(v) = O linear transformation T .

, L (Rⁿ,R^m) additive identity, zero mapping, kernel Rⁿ V ⊆ Rⁿ

zero mapping N(V ). T1, T2∈ N(V), r∈ R, v∈ V

(T1+ rT2)(v) = T1(v) + r(T2(v)) = O + rO = O.

V ⊆ ker(T1+ rT2), T1+ rT2∈ N(V). N(V ) L (Rⁿ,R^m) subspace.

Question 6.2. L (Rⁿ,R^m) Rⁿ R^m linear transformations vector

space. Rⁿ subspace V ,

S(V ) ={T ∈ L (Rⁿ,R^m)| ker(T) ⊆ V}.

S(V ) L (Rⁿ,R^m) subspace?

R^m subspace W ,

R(W ) ={T ∈ L (Rⁿ,R^m)| T(Rⁿ)⊆ W}.

R(W ) L (Rⁿ,R^m) subspace?

I(W ) ={T ∈ L (Rⁿ,R^m)| W ⊆ T(Rⁿ)}.

I(W ) L (Rⁿ,R^m) subspace?

6.2. Basis and Dimension

, vector space basis , finitely generated vector

space dimension 性 .

Rⁿ subspace , basis . basis, subspace

, basis 線性 . basis subspace

spanning vectors linearly independent ( Proposition 4.1.8).

vector space.

Definition 6.2.1. V vector space v₁, . . . , vn∈ V. c1, . . . , cn∈ R, c₁v₁+···+cnv_n v₁, . . . , v_n linear combination. v₁, . . . , v_n linear combination

, Span(v1, . . . , v_n) , Span(v1, . . . , v_n) ={∑ⁿi=1civ_i| c1, . . . , c_n∈ R}.

Span(v1, . . . , vn) V subspace ( Proposition 1.3.2).

v₁, . . . , v_n subspace. W V subspace v₁, . . . , v_n∈ W, subspace 數 性 Span(v1, . . . , v_n)⊆ W. , Span(v1, . . . , v_n) = V

, .

Definition 6.2.2. V vector space. v₁, . . . , vn∈ V Span(v1, . . . , vn) = V , {v1, . . . , v_n} V spanning set. V finitely generated vector space.

finitely generated vector space ? vector space

線性 . Rⁿ finitely generated ( e₁, . . . en

). 初 Rⁿ subspaces, finitely generated vector space

. vector space, finitely generated,

, .

Example 6.2.3. 前 vector space finitely generated vector space.

(A) Mm×n finitely generated. M_{i, j}∈ Mm×n, (i, j)-th entry 1,

entry 0 m× n matrix. m× n matrix Mi, j 1≤ i ≤ m,

1≤ j ≤ n linear combination. {Mi, j| 1 ≤ i ≤ m,1 ≤ j ≤ n} Mm×n spanning set, Mm×n finitely generated vector space.

(B) P(R) finitely generated vector space. { f1(x), . . . , f_n(x)} P(R) spanning set, f1(x), . . . , f_n(x) m, f1(x), . . . , f_n(x) linear combi- nation c₁f₁(x) +··· + cnf_n(x) 數 大 m. Span( f₁(x), . . . , f_n(x))

數大 m . { f1(x), . . . , f_n(x)} P(R) spanning set ,

P(R) finitely generated. 數 n Pn(R)

finitely generated vector space. {xⁿ, . . . , x, 1} P_n(R) spanning set.

大 finite generated vector space subspace finitely gener-

ated. , (大 ).

linearly independence , .

Spanning set linear combination 性, linear independence

linear combination 性. Rⁿ linearly independent

vector space.

Definition 6.2.4. V vector space v₁, . . . , v_n∈ V. c₁, . . . , c_n∈ R 0 c1v₁+···+cnv_n= O, v₁, . . . , vn linearly dependent. , c1, . . . , cn

0 c₁v₁+··· + cnv_n= O, v₁, . . . , v_n linearly independent.

Rⁿ , v₁, . . . , v_n linearly independent, c₁, . . . , c_n c1v₁+··· + cnv_n. v₁, . . . , vn linear combination

. , v₁, . . . , v_n linearly dependent, d₁, . . . , d_n 0 d1v₁+···+cnv_n= O. c1, . . . , c_n∈ R, c1v₁+···+cnv_n (c₁+ d₁)v₁+···+(cn+ d_n)v_n

v₁, . . . , vn linear combination, .

(c₁+ d₁)v₁+··· + (cn+ d_n)v_n=

(c1v₁+··· + cnv_n) + (d1v₁+··· + dnv_n) = c1v₁+··· + cnv_n+ O = c1v₁+··· + cnv_n. Example 6.2.5. P(R) xⁿ, xⁿ⁻¹, . . . , x, 1 linearly independent.

cn, . . . , c1, c0 ∈ R cnxⁿ+··· + c1x + c01 , ,

c_n, . . . , c₁, c₀ 0. P(R) linearly independent

, Lagrange interpolation polynomials. , 大

.

a, b, c 數, p₁(x), p₂(x), p₃(x)

p1(a) = 1, p₁(b) = p₁(c) = 0, p2(b) = 1, p₂(a) = p₂(c) = 0 and p3(c) = 1, p₃(a) = p₃(b) = 0.

p1(b) = p1(c) = 0, p1(x) (x− b)(x − c) , 數 r p₁(x) = r(x− b)(x − c). p₁(a) = 1, 代 x = a r = 1/(a− b)(a − c).

p2(x), p3(x)

p1(x) = (x− b)(x − c)

(a− b)(a − c), p2(x) = (x− a)(x − c)

(b− a)(b − c) and p3(x) = (x− a)(x − b) (c− a)(c − b).

p1(x), p2(x), p3(x) linearly independent. , f (x) = c1p1(x) + c₂p₂(x) + c₃p₃(x), 代 x = a p₁(a) = 1, p₂(a) = p₃(a) = 0, f (a) = c₁.

f (b) = c2, f (c) = c3. f (x) , f (a) = f (b) = f (c) = 0, c1= c2= c₃= 0. c₁= c₂= c₃= 0 c₁p₁(x) + c₂p₂(x) + c₃p₃(x)

, p1(x), p₂(x), p₃(x) linearly independent.

. n 數 a₁, . . . , a_n, n

n− 1 p1(x), . . . , pn(x) pi(ai) = 1 j̸= i , pi(aj) = 0. 前 , p₁(x), . . . , p_n(x) linearly independent.

Spanning set linearly independent 性 ,

. .

Lemma 6.2.6. V vector space v₁, . . . , vn∈ V.

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

(2) v₁, . . . , v_n−1 linearly independent, v₁, . . . , v_n−1, v_n linearly indepen- dent v_n̸∈ Span(v1, . . . , v_n−1).

(3) w₁, . . . , w_m∈ V. w₁, . . . , w_m∈ Span(v1, . . . , v_n) m > n, w₁, . . . , w_m linearly dependent.

Proof. (1), (2), (3) Lemma 4.2.1, Lemma 4.2.4 Lemma 4.2.5 vector space . (1), (2) Lemma 4.2.1, Lemma 4.2.4 .

(3) . Lemma 4.2.5 R^m ,

vector space, .

w₁, . . . , wm∈ Span(v1, . . . , vn), j = 1, . . . , m, wj v₁, . . . , vn

linear combination. , a_{1, j}, . . . , a_{i, j}, . . . , a_{n, j}∈ R w_j= a1, jv₁+··· + ai, jv_i+··· + an, jv_n.

c1, . . . , c_m∈ R 0 c1w₁+··· + cmw_m= O, w₁, . . . , w_m linearly dependent. c1w₁+···+cmw_m w_j v₁, . . . , vn linear combination

(c1a1,1+··· + cma1,m)v1+··· + (c1ai,1+··· + cmai,m)vi+··· + (c1am,1+··· + cmam,m)vm. (6.1)

c1, . . . , c_m ∈ R (6.1) v_i 數 0,

c1w₁+··· + cmw_m= O.

















a_1,1x₁+··· + a1,mx_m = 0 ...

a_i,1x₁+··· + ai,mx_m = 0 ...

a_n,1x₁+··· + an,mx_m = 0

x₁= c₁, . . . , x_m= c_m, c₁w₁+··· + cmw_m= O. homogeneous

linear system 數 n 數 數 m, Corollary 3.4.7

0 c₁, . . . , c_m∈ R x₁= c₁, . . . , x_m= c_m . w₁, . . . , w_m linearly

dependent. 

finitely generated vector space subspace finitely generated.

Proposition 6.2.7. V finitely generated vector space. W V subspace, W finitely generated vector space.

Proof. V finitely generated , v₁, . . . , v_n∈ V Span(v₁, . . . , v_n) = V .

{O} = Span(O) finitely generated, W̸= {O} . ,

W finitely generated. w₁∈ W w₁̸= O. W finitely generated, Span(w₁)̸= W, w₂∈ W w₂̸∈ Span(w1). Lemma 6.2.6 (2) w₁, w₂ linearly independent. , W finitely generated, Span(w1, w2)̸= W

w₃∈ W w₃̸∈ Span(w1, w₂). Lemma 6.2.6 (2) w₁, w₂, w₃ linearly independent.

, 數學 w₁, . . . , w_k∈ W linearly independent.

Span(w1, . . . , w_k)̸= W, w_k+1∈ W w_k+1̸∈ Span(w1, . . . , w_k). Lemma 6.2.6 (2) w₁, . . . , w_k, w_k+ linearly independent. 數學 , W finitely generated, m∈ N, w₁, . . . , wm∈ W linearly independent. m > n

. w₁, . . . , w_m∈ W ⊆ V = Span(v1, . . . , v_n), Lemma 6.2.6 (3) w₁, . . . , w_m linearly dependent. W finitely generated vector space.  Question 6.3. P(R) n∈ N, xⁿ, xⁿ⁻¹, . . . , x, 1 linearly independent,

P(R) finitely generated vector space.

spanning set 性 linearly independent 性, {v1, . . . , v_n} vector space V spanning set linearly independent, V

v₁, . . . , v_n linear combination, .

Definition 6.2.8. V vector space. {v1, . . . , vn} V spanning set linearly independent, v₁, . . . , v_n V basis.

finitely generated vector space basis. 性 finitely generated vector space ,

vector space finitely generated, .

Theorem 6.2.9. V ̸= {O} finitely generated vector space. v₁, . . . , vn∈ V V basis. w₁, . . . , w_m∈ V V basis, m = n.

Proof. Theorem 4.3.1 Theorem 4.3.2 vector space .

Theorem 4.3.1 basis 性. finitely generated

性 , 數學 .

vector space spanning set 數 數學 . V

. V = Span(u). V ̸= {O}, u̸= O. u linearly

independent {u} V spanning set u V basis. vector

space k , basis . V k + 1 vector

space, V = Span(u₁, . . . , u_k, u_k+1). W = Span(u₁, . . . , u_k). W

k vector space, v₁, . . . , v_n∈ W W basis.

Span(v1, . . . , vn) = W v₁, . . . , vn linearly independent. W = V , v₁, . . . , vn

V basis. W̸= V, u_k+1̸∈ W ( V = Span(u₁, . . . , u_k, u_k+1)⊆ W

). u_k+1̸∈ W = Span(v1, . . . , v_n) Lemma 6.2.6 (2) v₁, . . . , v_n, u_k+1 linearly independent. V = Span(u1, . . . , u_k, u_k+1) Span(u1, . . . , u_k) = W = Span(v₁, . . . , v_n)

V = Span(v1, . . . , vn, u_k+1). v₁, . . . , vn, u_k+1 V basis.

basis 數 , Lemma 4.2.5 Theorem 4.3.2

vector space , R^m . ,

. 

Theorem 6.2.9 V basis 數 . n

V basis, V basis n .

.

Definition 6.2.10. V finitely generated vector space. V basis

數 V dimension ( ), dim(V ) .

finitely generated vector space basis 數 ,

finitely generated vector space finite dimensional vector space.

Example 6.2.11. Example 6.2.3 finite dimensional vector space .

(A) Example 6.2.3 (A) {Mi, j∈ Mm×n| 1 ≤ i ≤ m,1 ≤ j ≤ n} Mm×n

spanning set. linearly independent, {Mi, j| 1 ≤ i ≤ m,1 ≤ j ≤ n}

Mm×n basis. dim(Mm×n) = m× n.

(B) {xⁿ, . . . , x, 1} Pn(R) spanning set linearly independent.

xⁿ, . . . , x, 1 P_n(R) basis, dim(P_n(R)) = n + 1.

finite dimensional vector space dimension 性 , .

Proposition 4.3.4, Corollary 4.3.6.

Proposition 6.2.12. V finite dimensional vector space.

(1) {v1, . . . , v_n} V spanning set, dim(V )≤ n. , v₁, . . . , v_n linearly dependent, dim(V ) < n.

(2) v₁, . . . , vn∈V linearly independent, dim(V )≥ n. , {v1, . . . , vn} V spanning set, dim(V ) > n.

(3) W V subspace, dim(W )≤ dim(V). dim(W ) = dim(V ) V = W .

(4) v₁, . . . , vn∈ V. .

(a) v₁, . . . , v_n V basis.

(b) dim(V ) = n {v1, . . . , vn} V spanning set.

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

, Proposition 6.2.12 (4) v₁, . . . , v_n V basis ,

dim(V ) n, spanning set linearly independent

.

Example 6.2.13. Example 6.2.5 a, b, c 數, p₁(x) = (x− b)(x − c)

(a− b)(a − c), p₂(x) = (x− a)(x − c)

(b− a)(b − c) and p₃(x) = (x− a)(x − b) (c− a)(c − b).

p1(a) = 1, p1(b) = p1(c) = 0; p2(b) = 1, p2(a) = p2(c) = 0; p3(c) = 1, p3(a) = p3(b) = 0.

p1(x), p₂(x), p₃(x)∈ P2(R) linearly independent dim(P2(R)) = 3, Propo- sition 6.2.12 (4) p₁(x), p₂(x), p₃(x) P₂(R) basis.

n 數 a₁, . . . , an, n n−1 p1(x), . . . , pn(x) p_i(a_i) = 1 j̸= i , pi(a_j) = 0. p₁(x), . . . , p_n(x)∈ Pn−1(R) linearly independent,

dim(Pn−1(R)) = n p1(x), . . . , pn(x) Pn−1(R) basis.

6.3. Linear Transformation

vector space linear transformation 性 . Rⁿ

, vector spaces , 數 vector space

數 , .

Definition 6.3.1. V,W vector spaces, T : V → W 數. T v₁, . . . , v_k∈ V c₁, . . . , c_k∈ R

T (c1v₁+··· + ckv_k) = c1T (v1) +··· + ckT (vk).

T linear transformation. T linear.

c1v₁+··· + cnv₁ V linear combination, c1T (v1) +··· + ckT (v_k)

W linear combination, . O∈ V V zero element ,

linear transformation , T (O) = T (O + O) = T (O) + T (O). T (O)

, T (O) W zero element. linear transformation

T : V→ W, V zero element W zero element. V̸= W ,

zero element , O , .

T (O) = O .

T : V→ W linear transformation, V linear

combination 代 T linear transformation . Rⁿ (

Proposition 5.1.3), linear combination .

Proposition 6.3.2. V,W vector spaces T : V → W 數. T linear transformation u, v∈ V, r ∈ R T (u + rv) = T (u) + rT (v).

T₁, T₂ V W linear transformation , T₁, T₂

T1+ T₂: V → W, (T₁+ T₂)(v) = T₁(v) + T₂(v), ∀v ∈ V. linear transformation T : V → W 數 . r∈ R, rT : V → W (rT )(v) = r(T (v)), ∀v ∈ V.

T₁+ T₂ rT V W linear transformation ( Proposition 5.1.6).

數 vector space 8 , (3) additive

identity V W zero mapping ( V W O).

( ).

Proposition 6.3.3. V,W vector spaces, L (V,W) V W linear

transformation . 數 數 , L (V,W) vector

space.

linear transformation “ 數”. U,V,W vector

spaces, T : V → W T^′: W → U linear transformations, T^′◦ T : V → U T^′◦ T(v) = T^′(T (v)), ∀v ∈ V. ( Proposition 5.1.7).

Proposition 6.3.4. U,V,W vector spaces. T : V → W T^′: W → U linear transformations, T^′◦ T : V → U linear transformation.

Rⁿ R^m linear transformation standard matrix representation

linear transformation basis . linear

transformation ( Theorem 5.1.8).

Theorem 6.3.5. V,W vector spaces v₁, . . . , vn∈ V, V basis.

w₁, . . . , w_n∈ W, linear transformation T : V → W, i = 1, . . . , n T (vi) = w_i.

linear transformation 數 性, subspace .

T : V → W linear transformation , V subspace V^′ T (V^′) ={T(v) ∈ W | v ∈ V^′} = {w ∈ W | w = T(v), for some v ∈ V^′}.

W^′ W subspace,

T⁻¹(W^′) ={v ∈ V | T(v) ∈ W^′}.

( Proposition 5.2.1).

Proposition 6.3.6. V,W vector spaces T : V → W linear transformation.

V^′ V subspaces, T (V^′) W subspace. , W^′ W subspaces,

T⁻¹(W^′) V subspace.

, V^′= V W^′={O} ,

T (V ) ={w ∈ W | w = T(v) for some v ∈ V} and T⁻¹({O}) = {v ∈ V | T(v) = O}

subspaces, T linear transformation . Rⁿ .

Definition 6.3.7. V,W vector spaces T : V → W linear transformation.

W subspace T (V ) T range ( image). V subspace T⁻¹({O})

T kernel, ker(T ) .

, linear transformation T : V → W onto range T (V ) W .

T range ? ( Proposition 5.2.4).

Proposition 6.3.8. V,W vector spaces T : V → W linear transformation.

{v1, . . . , v_n} V spanning set.

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

, T onto W = Span(T (v₁), . . . , T (v_n)).

kernel one-to-one . Proposition 5.2.6

.

Proposition 6.3.9. V,W vector spaces T : V → W linear transformation.

T one-to-one ker(T ) ={O}.

Proposition 6.3.8 T : V→ W onto spanning set. T

V spanning set W spanning set. linear transformation

linearly independent ? .

Proposition 6.3.10. V,W vector spaces T : V → W one-to-one linear transformation. v₁, . . . , v_n∈ V linearly independent T (v1), . . . , T (v_n)∈ W linearly independent.

Proof. (⇒) v₁, . . . , vn∈ V linearly independent, T (v1), . . . , T (vn)∈ W linearly independent. , T (v₁), . . . , T (v_n)∈ W linearly dependent,

c1, . . . , cn∈ R 0 c1T (v1) +··· + cnT (vn) = O. T linear, O = c₁T (v₁) +··· + cnT (v_n) = T (c₁v₁+··· + cnv_n). T one-to-one, c1v₁+··· + cnv_n= O T (c1v₁+··· + cnv_n) = O. v₁, . . . , v_n∈ V linearly independent c1=··· = cn= 0. c1, . . . , cn∈ R 0 ,

T (v₁), . . . , T (v_n)∈ W linearly independent.

(⇐) T one-to-one, T linear transformation.

T (v1), . . . , T (v_n)∈ W linearly independent. v₁, . . . , v_n∈ V linearly dependent, c₁, . . . , c_n∈ R 0 c₁v₁+···+cnv_n= O. T linear,

c₁T (v₁) +··· + cnT (v_n) = T (c₁v₁+··· + cnv_n) = T (O) = O. T (v₁), . . . , T (v_n)∈ W linearly independent, c1=··· = cn= 0 . v₁, . . . , vn∈ V linearly

independent. 

T : V → W one-to-one onto , invertible, T⁻¹: W→ V, v∈ V T⁻¹◦ T(v) = T⁻¹(T (v)) = v w∈ W T◦ T⁻¹(w) =

T (T⁻¹(w)) = w. T⁻¹ T inverse. Theorem 5.3.12 Rⁿ

linear transformation invertible, inverse linear transformation.

, Theorem 5.3.12 standard matrix representation

. matrix representation, .

Theorem 6.3.11. V,W vector spaces T : V→ W linear transformation.

T invertible, T inverse T⁻¹: W → V linear transformation.

Proof. w∈ W, T one-to-one onto, v∈ V T (v) = w.

數 T⁻¹(w) = v. T⁻¹ W V 數.

T⁻¹: W → V linear transformation. w₁, w₂∈ V, r ∈ R,

T⁻¹(w1+ rw2) = T⁻¹(w1) + rT⁻¹(w2). T⁻¹(w1) = v1, T⁻¹(w2) = v2. T (v1) = w1

T (v₂) = w₂. T⁻¹(w₁+ rw₂) T⁻¹(w₁) + rT⁻¹(w₂) = v₁+ rv₂

T (v1+ rv₂) = w₁+ rw₂. T linear transformation, T (v1+ rv₂) = T (v1) + rT (v2) = w1+ rw2. T⁻¹(w1+ rw2) = v1+ rv2= T⁻¹(w1) + rT⁻¹(w2),

T⁻¹ linear transformation. 

T : V→ W invertible linear transformation , T isomor-

phism. V W vector space T

數. T onto, T V W spanning set, T one-to-one

T linearly independent , .

Theorem 6.3.12. V,W vector spaces T : V → W isomorphism. v₁, . . . , v_n V basis T (v1), . . . , T (vn) W basis.

Proof. (⇒) v₁, . . . , v_n V basis. T onto {v1, . . . , v_n} V spanning set, Proposition 6.3.8 {T(v1), . . . , T (vn)} W spanning set.

T one-to-one v₁, . . . , v_n ∈ V linearly independent, Proposition 6.3.10 T (v1), . . . , T (vn)∈ W linearly independent. T (v1), . . . , T (vn) W basis.

(⇐) T (v1), . . . , T (vn) W basis. T⁻¹: W→ V linear transformation (Theorem 6.3.11), T : V → W inverse, T⁻¹ one-to-one onto,

T⁻¹: W → V isomorphism. T (v1), . . . , T (v_n) W basis, 前

T⁻¹(T (v1)), . . . , T⁻¹(T (vn)) V basis. i = 1, . . . , n T⁻¹(T (vi)) = vi,

v₁, . . . , v_n V basis. 

V,W finite dimensional vector space , Theorem 6.3.12, T : V → W

isomorphism, V W basis 數 , dim(V ) = dim(W ).

dim(V )̸= dim(W), V W isomorphism. 性

, .

Corollary 6.3.13. V,W finite dimensional vector spaces. T : V → W isomorphism dim(V ) = dim(W ).

Proof. (⇒) v₁, . . . , v_n V basis, dim(V ) = n. T : V → W isomorphism Theorem 6.3.12 T (v1), . . . , T (v_n) W basis. dim(W ) = n = dim(V ).

(⇐) dim(V ) = dim(W ), v₁, . . . , vn w₁, . . . , wn V basis W basis. linear transformation T : V→W T (v_i) = w_i,∀i = 1,...,n ( Theo- rem 6.3.5). T isomorphism. 6.3.8 T (V ) = Span(T (v1), . . . , T (vn)) = Span(w₁, . . . , w_n). w₁, . . . , w_n W basis Span(w₁, . . . , w_n) = W .

T (V ) = W , T onto. v∈ ker(T). v₁, . . . , v_n V basis, c1, . . . , cn∈ R v = c₁v₁+··· + cnv_n. T (v) = O,

O = T (c₁v₁+··· + cnv_n) = c₁T (v₁) +··· + cnT (v_n) = c₁w₁+··· + cnw_n.

w₁, . . . , w_n linearly independent, c₁=··· = cn= 0. v = c₁v₁+···+cnv_n= O.

ker(T ) ={O}, Proposition 6.3.9 T one-to-one, T isomorphism.



6.4. Coordinatization

, linear transformation, vector space

. vector space , Rⁿ

.

V finite dimensional vector space, V basis, basis

, , basis, ordered

basis. , basis , ordered

basis.([ ordered basis , (v1, . . . , vn) , .

1 0 ]

, [0

1

]) ([

0 1 ]

, [1

0 ])

R² ordered basis.

, ordered basis , ordered

basis. B = (v1, . . . , v_n) V ordered basis, B

ordered basis (v1, . . . , vn). Rⁿ standard basis, E (e1, . . . , en) ordered basis.

vector space V ordered basis B = (v1, . . . , v_n) , V

“ ” (coordinatization). v∈ V, B ordered basis

v v = c₁v₁+··· + cnv_n ,



 c₁

... c_n



 B v .

, _B[v] B v . ,

B[v] Rⁿ . vector space ,

Rⁿ .

Example 6.4.1. 前 vector space .

(A) M2×2 ordered basis E =

([ 1 0 0 0

] ,

[ 0 1 0 0

] ,

[ 0 0 1 0

] ,

[ 0 0 0 1

])

( basis M2×2 standard basis). M2×2

[ a b c d

] , [ a b

c d ]

= a [ 1 0

0 0 ]

+ b [ 0 1

0 0 ]

+ c [ 0 0

1 0 ]

+ d [ 0 0

0 1 ]

, [ a b

c d ]

E

E

[ a b c d

]

=





 a b c d





.

E

[ 1 −2

−3 4

]

=





 1

−2−3 4





.

(B) P2(R) x², x, 1 basis standard basis. E = (x², x, 1) ordered basis. 2x²− 3x + 4 E



 2

−3 4



,

E[2x²− 3x + 4] =



 2

−3 4



.

ordered basis B = (p1(x), p2(x), p3(x))

p₁(x) =−(x − 1)(x + 1), p2(x) = (1/2)x(x + 1) and p₃(x) = (1/2)x(x− 1) ( Example 6.2.5, Example 6.2.13).

p₁(0) = 1, p₁(1) = p₁(−1) = 0; p2(1) = 1, p₂(0) = p₂(−1) = 0; p3(−1) = 1, p3(0) = p₃(1) = 0, 2x²−3x+4 = c1p1(x) + c2p2(x) + c3p3(x), 代 x = 0, 1,−1, c1= 4, c2= 3, c3= 9.

B[2x²− 3x + 4] =



4 3 9



.