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# Linear Transformation of General Vector

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. 學 大 大 ( ) 線性

, 學 , 線性代數

. , 線性代數

, .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

, ,

, . , 性

, . , .

v

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## Space

, 前 vector space. vector space

vector space linear transformation. 前 , vector space

6.1. Vector Space and Subspace

Rm , 線性 Rm ,

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

r∈ R v∈ V, r v rv V ( 數

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

(4)

(7) r∈ R u, v∈ V r(u + v) = ru + rv.

(8) u∈ V, 1u = u.

, vector space,

space + , 數 數 ,

vector space , 數 . vector space

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,

, 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.

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).

(5)

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) 1r(rv) =rrv =1v, 性 (8) 1v = v, 1r(rv) = v̸= O.

rv = O, Proposition 6.1.3 (3) 1r(rv) = 1rO = 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

.

space 數 . , 數 “ ” ,

w + (−v) w− v.

. 2u + v = w, 1/2 u = 12(w− v).

vector space .

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

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) = bmxm+··· + b1x + b0∈ P(R), m≤ n, g(x) g(x) = bnxn+···+bmxm+···+b1x + b0,

bn = bn−1 =··· = bm+1= 0. 數 ,

f (x) + g(x) =ni=0(ai+ bi)xi. r∈ R,r f (x) r f (x) =ni=0(rai)xi.

, 數 數 ,

P(R) . vectors

space 8 .

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

n i=0

(ai+ bi)xi=

### ∑

n i=0

(bi+ ai)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) =

n i=0

(ai+ bi)xi+

n i=0

cixi=

### ∑

n i=0

((ai+ bi) + ci)xi,

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

n i=0

aixi+

n i=0

(bi+ ci)xi=

### ∑

n i=0

(ai+ (bi+ ci))xi.

(ai+ bi) + ci= ai+ (bi+ ci), ( f (x) + g(x)) + h(x) = f (x) + (g(x) + h(x)).

(6)

(3) g(x) = 0 =ni=0bixi∈ P(R), bi= 0, ∀i = 0,1,...,n.

f (x) =ni=0aixi∈ P(R), f (x) + g(x) =

n i=0

(ai+ bi)xi=

### ∑

n i=0

aixi= f (x).

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

n i=0

(ai− ai)xi=

### ∑

n i=0

0xi= 0.

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

n i=0

(sai)xi) =

n i=0

(r(sai))xi=

n i=0

((rs)ai)xi= (rs)

### ∑

n i=0

aixi= (rs) f (x).

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

n i=0

((r + s)ai)xi=

n i=0

(rai+ sai)xi=

n i=0

(rai)xi+

### ∑

n i=0

(sai)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(

n i=0

(ai+ bi)xi) =

n i=0

(r(ai+ bi))xi=

### ∑

n i=0

(rai+ rbi)xi= r f (x) + rg(x).

(8) f (x) =ni=0aixi∈ P(R), 1 f (x) =

n i=0

(1ai)xi=

### ∑

n i=0

aixi= f (x).

P(R) 數 vector space 8 , 數

P(R) vector space.

(C) L (Rn,Rm) Rn Rm linear transformations .

L (Rn,Rm) 數 , 數 L (Rn,Rm) (

Proposition 5.1.6). linear transformation 數

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)).

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

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

(3) zero element zero mapping, T :Rn→ Rm, T (v) = O,

∀v ∈ Rn linear transformation. L (Rn,Rm) 數 vector space

8 , 數 L (Rn,Rm) vector space.

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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

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, VW 性, 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, AT= A. n× n O

symmetric matrix. A, B∈ Mn×n AT = A, BT = B, r ∈ R,

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

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

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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.

Pn(R) Pn(R) = {∑ni=0aixi| ai∈ R}. Pn(R)

( 數學 數 −∞, 0. 代數

, ). f (x) =ni=0aixi, g(x) =ni=0bixi∈ Pn(R), r∈ R, f (x) + rg(x) =ni=0(ai+ rbi)xi∈ Pn(R). Pn(R) P(R) subspace. ,

n , P(R) subspace .

. , 數 n

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

vector space.

(C) L (Rn,Rm) Rn Rm linear transformations vector space.

Rn subspace V ,

N(V ) ={T ∈ L (Rn,Rm)| V ⊆ ker(T)}.

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

, L (Rn,Rm) 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,Rm) subspace.

Question 6.2. L (Rn,Rm) Rn Rm linear transformations vector

space. Rn subspace V ,

S(V ) ={T ∈ L (Rn,Rm)| ker(T) ⊆ V}.

S(V ) L (Rn,Rm) subspace?

Rm subspace W ,

R(W ) ={T ∈ L (Rn,Rm)| T(Rn)⊆ W}.

R(W ) L (Rn,Rm) subspace?

I(W ) ={T ∈ L (Rn,Rm)| W ⊆ T(Rn)}.

I(W ) L (Rn,Rm) subspace?

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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+···+cnvn v1, . . . , vn linear combination. v1, . . . , vn linear combination

, Span(v1, . . . , vn) , Span(v1, . . . , vn) ={∑ni=1civi| c1, . . . , cn∈ R}.

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

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

, .

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

ﬁnitely generated vector space ? vector space

). 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. Mi, 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), . . . , fn(x)} P(R) spanning set, f1(x), . . . , fn(x) m, f1(x), . . . , fn(x) linear combi- nation c1f1(x) +··· + cnfn(x) 數 大 m. Span( f1(x), . . . , fn(x))

P(R) ﬁnitely generated. 數 n Pn(R)

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

(10)

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, . . . , vn∈ V. c1, . . . , cn∈ R 0 c1v1+···+cnvn= O, v1, . . . , vn linearly dependent. , c1, . . . , cn

0 c1v1+··· + cnvn= O, v1, . . . , vn linearly independent.

Rn , v1, . . . , vn linearly independent, c1, . . . , cn c1v1+··· + cnvn. v1, . . . , vn linear combination

. , v1, . . . , vn linearly dependent, d1, . . . , dn 0 d1v1+···+cnvn= O. c1, . . . , cn∈ R, c1v1+···+cnvn (c1+ d1)v1+···+(cn+ dn)vn

v1, . . . , vn linear combination, .

(c1+ d1)v1+··· + (cn+ dn)vn=

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

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

cn, . . . , 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,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),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, . . . , an, n

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

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Spanning set linearly independent 性 ,

. .

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

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

(2) v1, . . . , vn−1 linearly independent, v1, . . . , vn−1, vn linearly indepen- dent vn̸∈ Span(v1, . . . , vn−1).

(3) w1, . . . , wm∈ V. w1, . . . , wm∈ Span(v1, . . . , vn) m > n, w1, . . . , wm 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 Rm ,

vector space, .

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

linear combination. , a1, j, . . . , ai, j, . . . , an, j∈ R wj= a1, jv1+··· + ai, jvi+··· + an, jvn.

c1, . . . , cm∈ R 0 c1w1+··· + cmwm= O, w1, . . . , wm linearly dependent. c1w1+···+cmwm wj v1, . . . , vn linear combination

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

c1, . . . , cm ∈ R (6.1) vi 數 0,

c1w1+··· + cmwm= O.















a1,1x1+··· + a1,mxm = 0 ...

ai,1x1+··· + ai,mxm = 0 ...

an,1x1+··· + an,mxm = 0

x1= c1, . . . , xm= cm, c1w1+··· + cmwm= O. homogeneous

linear system 數 n數 m, Corollary 3.4.7

0 c1, . . . , cm∈ R x1= c1, . . . , xm= cm . w1, . . . , wm 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.

(12)

Proof. V ﬁnitely generated , v1, . . . , vn∈ V Span(v1, . . . , vn) = 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, . . . , wk∈ W linearly independent.

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

. w1, . . . , wm∈ W ⊆ V = Span(v1, . . . , vn), Lemma 6.2.6 (3) w1, . . . , wm 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, . . . , vn} vector space V spanning set linearly independent, V

v1, . . . , vn linear combination, .

Deﬁnition 6.2.8. V vector space. {v1, . . . , vn} V spanning set linearly independent, v1, . . . , vn 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, . . . , wm∈ V V basis, m = n.

Proof. Theorem 4.3.1 Theorem 4.3.2 vector space .

Theorem 4.3.1 basis 性. ﬁnitely generated

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, . . . , uk, uk+1). W = Span(u1, . . . , uk). W

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

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

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

(13)

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

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

basis 數 , Lemma 4.2.5 Theorem 4.3.2

vector space , Rm . ,

. 

Theorem 6.2.9 V basis 數 . n

V basis, V basis n .

.

Deﬁnition 6.2.10. V ﬁnitely generated vector space. V basis

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 Pn(R) basis, dim(Pn(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, . . . , vn} V spanning set, dim(V )≤ n. , v1, . . . , vn 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, . . . , vn V basis.

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

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

(14)

, Proposition 6.2.12 (4) v1, . . . , vn 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) pi(ai) = 1 j̸= i , pi(aj) = 0. p1(x), . . . , pn(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

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

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

T linear transformation. T linear.

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

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).

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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).

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 Rm 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, . . . , wn∈ W, linear transformation T : V → W, i = 1, . . . , n T (vi) = wi.

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}

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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, . . . , vn} V spanning set.

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

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

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, . . . , vn∈ V linearly independent T (v1), . . . , T (vn)∈ W linearly independent.

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

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

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

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

T (v1), . . . , T (vn)∈ W linearly independent. v1, . . . , vn∈ V linearly dependent, c1, . . . , cn∈ R 0 c1v1+···+cnvn= O. T linear,

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

independent. 

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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.

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

T linearly independent , .

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

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

T one-to-one v1, . . . , vn ∈ 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 (vn) W basis, 前

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

v1, . . . , vn 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

, .

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

Proof. (⇒) v1, . . . , vn V basis, dim(V ) = n. T : V → W isomorphism Theorem 6.3.12 T (v1), . . . , T (vn) 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 (vi) = wi,∀i = 1,...,n ( Theo- rem 6.3.5). T isomorphism. 6.3.8 T (V ) = Span(T (v1), . . . , T (vn)) = Span(w1, . . . , wn). w1, . . . , wn W basis Span(w1, . . . , wn) = W .

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

O = T (c1v1+··· + cnvn) = c1T (v1) +··· + cnT (vn) = c1w1+··· + cnwn.

w1, . . . , wn linearly independent, c1=··· = cn= 0. v = c1v1+···+cnvn= 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, . . . , vn) V ordered basis, B

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

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

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

v v = c1v1+··· + cnvn ,

 c1

... cn

 B v .

, B[v] B v . ,

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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

.

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