大學線性代數初步

大學線性代數初步

大學數學

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代數 . 線性代數 .

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v

Chapter 5

Linear Transformations of R ⁿ

Rⁿ 數, linear transformation.

linear transformation 性 . , linear transformation

.

5.1. Basic Properties

數學 , 數 . 線性代數 ,

vector space, linear transformation vector

spaces 數 .

5.1.1. Function. Rⁿ,R^m. Rⁿ R^m ,

v∈ Rⁿ, v R^m w. v∈ Rⁿ, w

T (v) , T :Rⁿ→ R^m, T (v) = w∈ R^m, ∀v ∈ Rⁿ

, T Rⁿ R^m function ( 數). Rⁿ T domain ( ),

R^m T codomain ( ). 數 , T :Rⁿ→ R^m 數,

v∈ Rⁿ, T (v) R^m . T (v)∈ R^m,

T (v) = w, T (v) = w^′, w̸= w^′, .

數 , . , 數

well-defined.

, 數 .

T :Rⁿ→ R^m 數, Rⁿ v̸= v^′, T (v) = T (v^′).

, Rⁿ v̸= v^′,

T (v) T (v^′) ( T (v)̸= T(v^′)), 數 ,

數 one-to-one ( ), injective. , 數

. T :Rⁿ→ R^m 數, R^m

91

w, Rⁿ w ( v∈ Rⁿ T (v) = w).

, R^m w, v∈ Rⁿ T (v) = w,

數 , 數 onto ( ), surjective.

數 one-to-one onto ( bijective), .

數 數 數 ( 數

inverse ( 數)), 數 ( identity

function). 數 invertible ( 數).

5.1.2. Linear Transformation. Rⁿ , 數,

, Rⁿ . 數 ,

.

Definition 5.1.1. T :Rⁿ→ R^m 數, T v₁, . . . , v_k ∈ Rⁿ c₁, . . . , c_k∈ R

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

T linear transformation. T linear.

c1v₁+··· + cnv₁ Rⁿ 線性 , c1T (v1) +··· + ckT (vk) R^m

線性 , . n̸= m . O∈ Rⁿ , linear

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

, T (O) R^m . linear transformation T :Rⁿ→ R^m,

Rⁿ R^m . n̸= m , Rⁿ R^m

, O , . T (O) = O linear

transformation Rⁿ R^m . 性 ,

, 性 .

Lemma 5.1.2. T :Rⁿ→ R^m linear transformation. T Rⁿ

R^m , T (O) = O.

, O Rⁿ , R^m ,

.

T :Rⁿ→ R^m linear transformation, Rⁿ

線性 代 T linear transformation , .

, subspace ( Proposition 4.1.2), ,

線性 .

Proposition 5.1.3. T :Rⁿ→ R^m 數, T linear transformation u, v∈ Rⁿ, r∈ R T (u + rv) = T (u) + rT (v).

Proof. (⇒) : T linear , u, v∈ V, r ∈ R T (u + rv) = T (u) + rT (v).

5.1. Basic Properties 93

(⇐) : u, v∈ Rⁿ, r∈ R T (u + rv) = T (u) + rT (v) 性

v₁, . . . , v_k∈ Rⁿ c1, . . . , c_k ∈ R T (c1v₁+··· + ckv_k) = c₁T (v1) +··· + ckT (vk).

數 k 數學 . ( k = 1),

v₁∈ Rⁿ, c₁∈ R T (c₁v₁) = c₁T (v₁). u = O, v = v₁, r = c₁, Lemma 5.1.2, T (c1v₁) = T (u + rv) = T (u) + rT (v) = O + rT (v) = c1T (v1). k = 1

. k , v₁, . . . , v_k ∈ Rⁿ c₁, . . . , c_k ∈ R T (c1v₁+··· + ckv_k) = c₁T (v1) +··· + ckT (vk). v₁, . . . , v_k, v_k+1∈ Rⁿ c1, . . . , c_k, c_k+1∈ R T (c1v₁+···+ckv_k+ c_k+1v_k+1) = c1T (v1) +···+ckT (v_k) + c_k+1T (v_k+1).

u = c₁v₁+··· + ckv_k, v = v_k+1 r = c_k+1. T (u) = c1T (v1) +··· + ckT (vk),

T (c1v₁+··· + ckv_k+ c_k+1v_k+1) =

T (u + rv) = T (u) + rT (v) = c₁T (v₁) +··· + ckT (v_k) + c_k+1T (v_k+1).

數學 T linear transformation. 

Example 5.1.4. (1) T :R³→ R² T (



 x1

x2

x₃



) =[

x1+ x2

x₁− x3

]

. T

linear transformation. u =



a1

a₂ a₃



,v =



b1

b₂ b₃



 ∈ R³, r∈ R. u+rv =



a1+ rb1

a₂+ rb₂ a₃+ rb₃



.

T ,

T (u + rv) = T (



a1+ rb1

a2+ rb2

a₃+ rb₃



) =[

(a1+ rb1) + (a2+ rb2) (a1+ rb1)− (a3+ rb3) ]

=

[a1+ a2+ rb1+ rb2

a1− a3+ rb1− rb3

] .

T (u) = T (



a₁ a2

a3



) =[ a1+ a₂ a1− a3

]

, T (v) = T (



b₁ b2

b3



) =[ b1+ b₂ b1− b3

] ,

T (u) + rT (v) =

[a₁+ a₂ a₁− a3

] + r

[b₁+ b₂ b₁− b3

]

=

[a₁+ a₂+ rb₁+ rb₂ a₁− a3+ rb₁− rb3

] .

T (u + rv) = T (u) + rT (v), T linear transformation.

(2) T :R³→ R² T (



 x1

x₂ x₃



) =[

x₁+ x₂+ 1 x₁− x3

]

. T linear

transformation. T , T (O) =

[1 0 ]

̸= O, Lemma 5.1.2 , T linear transformation.

(3) T :R³→ R² T (



 x₁ x2

x3



) =[

x²₁+ x₂ x1− x3

]

. T linear transfor-

mation. T (O) = O, T (



1 0 0



) =[ 1 1 ]

, T (



2 0 0



) =[ 4 2 ]

T (



2 0 0



) = T(2



1 0 0



) ̸= 2T(



1 0 0



).

linear transformation , T linear transformation.

linear transformation, Rⁿ

R^m linear transformation .

Lemma 5.1.5. A m× n matrix. T :Rⁿ→ R^m : T (v) = Av, ∀v ∈ Rⁿ. T linear transformation.

Proof. T well-defined, T Rⁿ R^m 數.

v∈ Rⁿ, T (v) = Av. A m× n matrix, Av m× 1

matrix ( column vector, v∈ Rⁿ n× 1 matrix), Av∈ R^m.

T Rⁿ R^m function.

T linear, u, v∈ Rⁿ r ∈ R, T (u + rv) =

T (u) + rT (v). T T (u) = Au, T (v) = Av, T (u + rv) = A(u + rv).

(Proposition 3.1.8 Proposition 3.1.9)

T (u + rv) = A(u + rv) = Au + A(rv) = Au + rAv = T (u) + rT (v).



Lemma 5.1.5 linear transformations. ,

linear transformations linear transformations. T1, T2 Rⁿ R^m linear transformation, T₁, T₂ linear transformation, T₁+ T₂.

前 , 數 . T1+ T2

Rⁿ R^m 數. v∈ Rⁿ, (T₁+ T₂)(v) = T₁(v) + T₂(v). ,

T₁+ T₂ Rⁿ R^m . , v∈ Rⁿ, T₁(v)∈ R^m

T2(v)∈ R^m, (T1+ T2)(v) = T1(v) + T2(v)∈ R^m. T1+ T2:Rⁿ→ R^m well-defined function. T₁:Rⁿ→ R^m, T₂:Rⁿ→ R^m linear transformation, T1+ T2:Rⁿ→ R^m linear transformation. u, v∈ Rⁿ

r∈ R, (T₁+ T₂)(u + rv) = (T₁+ T₂)(u) + r(T₁+ T₂)(v).

(T1+ T2)(u + rv) = T1(u + rv) + T2(u + rv).

T1, T₂ linear

T₁(u + rv) + T₂(u + rv) = T₁(u) + rT₁(v) + T₂(u) + rT₂(v).

5.1. Basic Properties 95

,

(T₁+ T₂)(u) + r(T₁+ T₂)(v) = T₁(u) + T₂(u) + r(T₁(v) + T₂(v)),

性 , (T₁+ T₂)(u + rv) = (T₁+ T₂)(u) + r(T₁+ T₂)(v), T₁+ T₂ Rⁿ R^m linear transformation.

Question 5.1. A1, A₂ m× n matrix. T1:Rⁿ→ R^m, T2:Rⁿ→ R^m, T₁(v) = A₁v, T₂(v) = A₂v, ∀v ∈ Rⁿ. T₁+ T₂ 數?

linear transformation T :Rⁿ→ R^m, c∈ R, 數 cT :Rⁿ→

R^m, (cT )(v) = c(T (v)), ∀v ∈ Rⁿ ( Rⁿ v

c T (v)). rT :Rⁿ→ R^m function. , linear

transformation. u, v∈ Rⁿ r∈ R,

(cT )(u + rv) = c(T (u + rv)) = cT (u) + rcT (v).

(cT )(u) + r(cT )(v) = c(T (u)) + rc(T (v)), (cT )(u + rv) = (cT )(u) + r(cT )(v), cT :Rⁿ→ R^m linear transformation.

Question 5.2. A m× n matrix. T :Rⁿ→ R^m T (v) = Av, ∀v ∈ Rⁿ.

c∈ R, cT 數?

T₁,··· ,Tk Rⁿ R^m linear transformations. c₁, . . . , c_k∈ R, 前 c₁T₁, . . . , c_kT_k Rⁿ R^m linear transformations. c1T2+ c₂T2 linear transformation.

數學 , c₁T₁+··· + ckT_k linear transformation. . Proposition 5.1.6. T₁,··· ,Tk Rⁿ R^m linear transformations, c₁, . . . , c_k∈ R c1T1+··· + ckTk Rⁿ R^m linear transformation.

linear transformation “ 數”. T :Rⁿ→ R^m

T^′:R^m→ R^k 數, v∈ Rⁿ, T (v)∈ R^m, T (v) T^′

. T (v) 代 T^′ , T^′(T (v))∈ R^k.

Rⁿ R^k 數, T, T^′ composite function ( 數), T^′◦T

. T^′◦ T : Rⁿ→ R^k T^′◦ T(v) = T^′(T (v)),∀v ∈ Rⁿ. . Proposition 5.1.7. T :Rⁿ→ R^m T^′:R^m→ R^k linear transformation, T^′◦T : Rⁿ→ R^k linear transformation.

Proof. T^′◦ T function, T^′◦ T linear, u, v∈ Rⁿ r∈ R, (T^′◦ T)(u + rv) = (T^′◦ T)(u) + r(T^′◦ T)(v). (T^′◦ T)(u + rv) = T^′(T (u + rv)). T , T^′ linear,

T^′(T (u + rv)) = T^′(T (u) + rT (v)) = T^′(T (u)) + rT^′(T (v)).

T^′(T (u)) = (T^′◦T)(u) T^′(T (v)) = (T^′◦T)(v) T^′◦T linear transformation.



Question 5.3. A m× n matrix, B k× m matrix. T :Rⁿ→ R^m, T^′:R^m→ R^k, T (v) = Av, ∀v ∈ Rⁿ T^′(w) = Bw, ∀w ∈ R^m. T^′◦ T 數?

v₁, . . . , vn Rⁿ basis , v∈ Rⁿ, c1, . . . , cn∈ R, v = c₁v₁+···+cnv_n. T :Rⁿ→ R^m linear transformation, linear transformation

, T (v) = c1T (v1) +··· + cnT (vn). , T (v1), . . . , T (vn)

R^m , v∈ Rⁿ, T (v) . .

Theorem 5.1.8. v₁, . . . , vn∈ Rⁿ, Rⁿ basis. w₁, . . . , wn∈ R^m, linear transformation T :Rⁿ→ R^m, T (v₁) = w₁, . . . , T (v_i) = w_i, . . . , T (v_n) = w_n. Proof. 性. T :Rⁿ→ R^m T (c₁v₁+··· + cnv_n) = c₁w₁+··· + cnw_n,

∀c1, . . . , cn∈ R. well-defined function. v∈ Rⁿ, T (v)

T (v)∈ R^m. v₁, . . . , v_n∈ Rⁿ, Rⁿ basis, c₁, . . . , c_n∈ R, v = c₁v₁+··· + cnv_n. T (v) = T (c1v₁+··· + cnv_n) = c1w₁+··· + cnw_n∈ R^m.

T linear transformation, u, v∈ Rⁿ r ∈ R,

T (u + rv) = T (u) + rT (v). v₁, . . . , v_n Rⁿ basis, c1, . . . , c_n d1, . . . , dn ∈ R u = c₁v₁+··· + cnv_n v = d₁v₁+··· + dnv_n. u + rv = (c₁+ rd₁)v₁+··· + (cn+ rd_n)v_n. T

T (u + rv) = (c1+ rd1)T (v1) +··· + (cn+ rdn)T (vn) = (c1+ rd1)w1+··· + (cn+ rdn)wn.

T (u) + rT (v) = T (c1v₁+··· + cnv_n) + rT (d1v₁+··· + dnv_n) =

(c₁w₁+··· + cnw_n) + r(d₁w₁+··· + dnw_n) = (c₁+ rd₁)w₁+··· + (cn+ rd_n)w_n. T (u + rv) = T (u) + rT (v).

性, . T^′:Rⁿ→ R^m linear transforma-

tion T^′(v1) = w1, . . . , T^′(vn) = wn, T^′̸= T, . , T^′̸= T v∈ Rⁿ T^′(v)̸= T(v). c₁, . . . , c_n, v = c₁v₁+··· + cnv_n, T, T^′

linear ,

T^′(v) = T^′(c₁v₁+··· + cnv_n) = c₁T^′(v₁) +··· + cnT^′(v_n) = c₁w₁+··· + cnw_n= T (v).

T^′(v)̸= T(v) , 性. 

Theorem 5.1.8 w₁, . . . , wn∈ R^m , basis

linear independent. , basis 性. Rⁿ

basis , basis R^m , Rⁿ

R^m linear transformation. , 數 ,

數 .

, , . linear

transformation , Theorem 5.1.8 linear transformations

5.2. Kernel and Range 97

basis , linear transformation

數.

5.2. Kernel and Range

Linear transformation linear combination ,

subspaces. linear transformation

subspaces, “kernel” “range”, subspace linear

transformation .

數 T :Rⁿ→ R^m, Rⁿ S,

T (S) ={T(v) ∈ R^m| v ∈ S} , S T R^m

. , R^m S^′,

T⁻¹(S^′) ={v ∈ Rⁿ| T(v) ∈ S^′} , S^′

. T (S) R^m , T⁻¹(S^′) Rⁿ

. w∈ T(S) , w S T ,

v∈ V w = T (v). T (S) T (S) ={w ∈ R^m| w = T(v), for some v ∈ S},

T (S) R^m , .

linear transformation , T :Rⁿ→ R^m linear transformation ,

V Rⁿ subspace .

T (V ) ={T(v) ∈ R^m| v ∈ V} = {w ∈ R^m| w = T(v), for some v ∈ V}

性. W R^m subspace,

T⁻¹(W ) ={v ∈ Rⁿ| T(v) ∈ W}

性. , .

Proposition 5.2.1. T :Rⁿ→ R^m linear transformation. V Rⁿ subspaces, T (V ) R^m subspace. , W R^m subspaces, T⁻¹(W ) Rⁿ subspace.

Proof. T (V ) R^m , T⁻¹(W ) Rⁿ .

subspaces, Proposition 4.1.2 , w, w^′∈ T(V) r∈ R, w + rw^′∈ T(V) v, v^′∈ T⁻¹(W ) r∈ R, v + rv^′∈ T⁻¹(W ).

, R^m w T (V ) , v∈ V

w = T (v). w, w^′∈ T(V), v, v^′∈ V T (v) = w, T (v^′) = w^′. r∈ R, w + rw^′= T (v) + rT (v^′). T linear, w + rw^′= T (v + rv^′). V Rⁿ subspace, v, v^′∈ V v + rv^′∈ V, w + rw^′= T (v + rv^′)∈ T(V).

, v, v^′∈ T⁻¹(W ), T (v), T (v^′)∈ W. r∈ R, T linear, T (v + rv^′) = T (v) + rT (v^′). W R^m subspace, T (v), T (v^′)∈ W T (v) + rT (v^′)∈ W. T (v + rv^′) = T (v) + rT (v^′)∈ W, v + rv^′∈ T⁻¹(W ). 

, V =Rⁿ W ={O} ,

T (Rⁿ) ={w ∈ R^m| w = T(v) for some v ∈ Rⁿ} and T⁻¹({O}) = {v ∈ Rⁿ| T(v) = O}

subspaces, T linear transformation .

subspace .

Definition 5.2.2. T :Rⁿ→ R^m linear transformation. R^m subspace T (Rⁿ) T range ( image). Rⁿ subspace T⁻¹({O}) T kernel,

ker(T ) .

linear transformation range. T :Rⁿ → R^m linear trans- formation. Proposition 5.2.1 T range T (Rⁿ) R^m subspace, dim(T (Rⁿ))≤ dim(R^m) = m. dim(T (Rⁿ)) = m, Corollary 4.3.6 T (Rⁿ) =R^m.

w∈ R^m, w∈ T(Rⁿ) v∈ Rⁿ w = T (v).

T onto. , T onto, w∈ R^m, v∈ Rⁿ

T (v) = w w∈ T(Rⁿ), R^m⊆ T(Rⁿ). T (Rⁿ)⊆ R^m, T (Rⁿ) =R^m. 性 .

Proposition 5.2.3. T :Rⁿ→ R^m linear transformation. T onto dim(T (Rⁿ)) = m.

, 數 onto range codomain

( ). 數 onto . Proposition 5.2.3

linear transformation, range dimension onto.

linear transformation range ? 性 .

Proposition 5.2.4. T :Rⁿ→ R^m linear transformation v₁, . . . , vn∈ Rⁿ Rⁿ spanning vectors.

T (Rⁿ) = Span(T (v₁), . . . , T (v_n)).

Proof. w∈ T(Rⁿ), v∈ Rⁿ w = T (v). v₁, . . . , v_n Rⁿ spanning vectors, c1,··· ,cn∈ R, v = c₁v₁+··· + cnv_n. T linear

w = T (v) = T (c₁v₁+··· + cnv_n) = c₁T (v1) +··· + cnT (vn)∈ Span(T(v1), . . . , T (v_n)), T (Rⁿ)⊆ Span(T(v1), . . . , T (vn)).

, w∈ Span(v1, . . . , v_n), c1,··· ,cn∈ R, w = c₁T (v1) +··· + cnT (vn). T linear

w = c₁T (v₁) +··· + cnT (v_n) = T (c₁v₁+··· + cnv_n)∈ T(Rⁿ),

Span(T (v₁), . . . , T (v_n))⊆ T(Rⁿ). T (Rⁿ) = Span(T (v₁), . . . , T (v_n)). 

5.2. Kernel and Range 99

Example 5.2.5. (1) T :R³→ R² T (



 x₁ x2

x3



) =[

x₁+ x₂ x1− x3

]

. Example 5.1.4 T linear transformation. R³ standard basis{e1, e₂, e₃}, T (



1 0 0



) =[ 1 1 ]

, T (



0 1 0



) =[ 1 0 ]

, T (



0 0 1



) =[ 0

−1 ]

.

[1 1 ]

, [1

0 ]

, [ 0

−1 ]

R²

spanning vectors, Proposition 5.2.4 T (R³) = Span(

[1 1 ]

, [1

0 ]

, [ 0

−1 ]

) =R². T onto.

(2) T :R²→ R³ T ( [ x₁

x₂ ]

) =



 x1

x₁+ x₂ x₁− x2



. T linear

transformation. R² standard basis{e1, e₂}, T ( [1

0 ]

) =



1 1 1



, T([ 0 1 ]

) =



 0 1

−1



. Proposition 5.2.4 T (R²) = Span(



1 1 1



,



 0 1

−1



).



1 1 1



,



 0 1

−1



 linearly

independent, dim(T (R²)) = 2. Proposition 5.2.3 T onto.

Question 5.4. T :Rⁿ→ R^m linear transformation. m > n, T onto

?

linear transformation T range {O}. T

O, T (v) = O, ∀v ∈ Rⁿ. linear transformation, , zero mapping.

Question 5.5. T :Rⁿ→ R^m linear transformation v₁, . . . , v_n∈ Rⁿ Rⁿ basis. T zero mapping T (v₁) =··· = T(vn) = O.

kernel linear transformation . T :Rⁿ→ R^m linear

transformation. T one-to-one, T (O) = O, v

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

.

Proposition 5.2.6. T :Rⁿ→ R^m linear transformation. T one-to-one dim(ker(T )) = 0, ker(T ) ={O}.

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

dim(ker(T )) = 0. , dim(ker(T )) = 0, ker(T ) ={O}, T one-to-one, v, v^′∈ Rⁿ v̸= v^′ T (v) = T (v^′). T linear, T (v−v^′) = T (v)−T(v^′) = O, v− v^′∈ T⁻¹({O}) = ker(T) = {O}. v = v^′ , T one-to-one.  Example 5.2.7. Example 5.2.5 linear transformation one-to-one.

(1) T :R³→ R² T (



 x₁ x2

x3



) =[

x₁+ x₂ x1− x3

]

. v =



a b c



 ∈ ker(T),

T (v) = T (



a b c



) =[ a + b a− c ]

= [0

0 ]

, a + b = 0 a− c = 0,



 1

−1 1



 ∈ ker(T),

ker(T )̸= {O} ( ker(T ) = Span(



 1

−1 1



)). T one-to-one.

(2) T :R²→ R³ T ( [ x1

x₂ ]

) =



 x1

x₁+ x₂ x₁− x2



. v = [a

b ]

∈ ker(T), T (v) =

T ( [a

b ]

) =



 a a + b a− b



 =



0 0 0



. a = 0, a + b = 0 a− b = 0, a = b = 0.

ker(T ) ={O}, Proposition 5.2.6 T one-to-one.

Proposition 5.2.6 T linear transformation . f (x) = x²

, f⁻¹(0) ={0} ( x = 0 x²= 0) f (x) (

f (1) = f (−1) = 1). f (x) linear. f linear transformation

, f⁻¹({0}) one-to-one.

Question 5.6. Proposition 5.2.6 T linear ? T

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

Question 5.7. T :Rⁿ→ R^m linear transformation. Question 5.5 T

zero mapping T range . T zero mapping T kernel

?

數, one-to-one , T linear transformation

, Proposition 5.2.6 T one-to-one.

O . linear transformation kernel

. , kernel .

5.3. Matrix Representation

m×n matrix A, 前 linear transformation T :Rⁿ→

R^m, T (v) = Av, ∀v ∈ Rⁿ. , Rⁿ R^m linear

transformations . linear transformation matrix

, 性 .

前 Theorem 5.1.8 linear transformation, linear trans-

formation Rⁿ basis , linear transforma-

tion. Rⁿ , basis, standard basis {e1, . . . , e_n}. T :Rⁿ→ R^m

5.3. Matrix Representation 101

linear transformation, 前 , T (e₁), . . . , T (e_n) R^m

vectors, v∈ Rⁿ, T (v) .

v∈ Rⁿ, c₁, . . . , c_n∈ R v = c₁e₁+··· + cne_n,

v v =





 c₁ c₂ ... c_n





. T linear, T (v) = T (c1e₁+··· + cne_n) =

c₁T (e₁) +··· + cnT (e_n). m× n matrix A, A i-th column T (e_i),

Av =



   

T (e1) T (e2) ··· T(en)









 c₁ c2

... c_n





= c1T (e1) +··· + cnT (en) = T (v).

v∈ Rⁿ, T (v) = Av. T v A

linear transformation. .

Theorem 5.3.1. Rⁿ R^m function T . T linear transformation m× n matrix A T (v) = Av, ∀v ∈ Rⁿ. m× n matrix A , i = 1, . . . , n, A i-th column T (ei), {e1, . . . , e_n} Rⁿ standard basis.

Proof. Lemma 5.1.5 , T (v) = Av, ∀v ∈ Rⁿ, T linear transformation.

, T :Rⁿ→ R^m linear transformation, 前 , A i-th

column T (e_i) m× n matrix, 性 T (v) = Av,∀v ∈ Rⁿ.

B m× n matrix T (v) = Bv, i = 1, . . . , n, Be_i

B i-th column. Bei = T (e_i), B i-th column T (ei). B

column 前 A column , 性. 

Theorem 5.3.1 Rⁿ R^m linear transformations m× n

matrices ( 數 , ).

linear transformation m× n matrix , .

Definition 5.3.2. T :Rⁿ → R^m linear transformation {e1, . . . , e_n} Rⁿ standard basis. i = 1, . . . , n, i-th column T (ei) m× n matrix T standard matrix representation.

T standard matrix representation T , [T ]

T standard matrix representation. v∈ Rⁿ, T (v) = [T ]v.

Example 5.3.3. Example 5.2.5 linear transformation standard matrix representation.

(1) T :R³→ R² T (



 x₁ x2

x3



) =[

x₁+ x₂ x1− x3

] .

T (e1) = T (



1 0 0



) =[ 1 1 ]

, T (e2) = T (



0 1 0



) =[ 1 0 ]

, T (e3) = T (



0 0 1



) =[ 0

−1 ]

,

[T ] =



   

T (e1) T (e2) T (e3)



 =[

1 1 0

1 0 −1 ]

.

[T ]



x₁ x₂ x3



 =[

1 1 0

1 0 −1 ]x₁

x₂ x3



 =[

x₁+ x₂ x₁− x3

]

= T (



 x₁ x₂ x3



).

(2) T :R²→ R³ T ( [ x1

x₂ ]

) =



 x1

x₁+ x₂ x₁− x2



.

T (e₁) = T ( [1

0 ]

) =



1 1 1



,T(e₂) = T ( [0

1 ]

) =



 0 1

−1



,

[T ] =



  

T (e₁) T (e₂)  



 =



 1 0 1 1 1 −1



.

[T ] [x1

x2

]

=



 1 0 1 1 1 −1



[ x1

x2

]

=



 x1

x1+ x2

x₁− x2



 = T([ x1

x2

] ).

standard matrix representation, linear transformation range kernel.

Proposition 5.3.4. T :Rⁿ → R^m linear transformation [T ]∈ Mm×n

standard matrix representation. T range [T ] column space, T kernel [T ] nullspace.

Proof. e₁, . . . , en Rⁿ basis, Proposition 5.2.4 T range, T (Rⁿ) = Span(T (e₁), . . . , T (e_n)). T (e₁), . . . , T (e_n) [T ] n column,

Span(T (e1), . . . , T (en)) [T ] column space. T range [T ] column space.

, v∈ ker(T), T (v) = O. standard matrix representation T (v) = [T ]v, [T ]v = O, v [T ] nullspace. ker(T ) [T ] nullspace. , v [T ] nullspace, [T ]v = O, T (v) = O, v∈ ker(T).

[T ] nullspace ker(T ), T kernel [T ] nullspace. 

5.3. Matrix Representation 103

A column space , A rank, rank(A) .

A nullspace A nullity, null(A) ( Definition 2.3.1).

Proposition 5.3.4, T range rank([T ]), T kernel

null([T ]), .

Corollary 5.3.5. T :Rⁿ→ R^m linear transformation [T ]∈ Mm×n standard matrix representation.

dim(T (Rⁿ)) = rank([T ]) and dim(ker(T )) = null([T ]).

T range T rank, T kernel

T nullity. 步 linear transformation matrix ,

linear transformation Dimension Theorem.

Theorem 5.3.6 (Dimension Theorem for Linear Transformation). T :Rⁿ→ R^m linear transformation.

dim(T (Rⁿ)) + dim(ker(T )) = n.

Proof. T standard matrix representation [T ] m× n matrix, Theorem 4.4.13

rank([T ]) + null([T ]) = n. Corollary 5.3.5, . 

Example 5.3.7. Example 5.3.3 linear transformation standard matrix representation range kernel.

(1) T :R³→ R² T (



 x1

x₂ x₃



) =[

x₁+ x₂ x₁− x3

]

, standard matrix repre-

sentation [T ] =

[ 1 1 0 1 0 −1

]

. [T ] column space Span(

[1 1 ]

, [1

0 ]

, [ 0

−1 ]

) =R². Proposition 5.3.4 T (R³) =R² ( Example 5.2.5(1) ). , [T ]

nullspace [T ]x = O,

{ x1 +x2 = 0

x₁ −x3 = 0 ∼

{ x1 +x2 = 0

−x2 −x3 = 0

. Proposition 5.3.4 ker(T ) = Span(



 1

−1 1



) ( Example 5.2.7(1)

). Dimension Theorem, dim(T (R³)) + dim(ker(T )) = 2 + 1 = 3.

(2) T :R²→ R³ T ( [ x1

x₂ ]

) =



 x1

x1+ x2

x₁− x2



, standard matrix represen-

tation [T ] =



 1 0 1 1 1 −1



. [T ] column space Span(



1 1 1



,



 0 1

−1



). Proposition

5.3.4 T (R³) = Span(



1 1 1



,



 0 1

−1



) ( Example 5.2.5(2) ). , [T ]

nullspace [T ]x = O,





x₁ = 0

x1 +x₂ = 0 x1 −x2 = 0

∼

{ x₁ = 0

x2 = 0 . Proposition 5.3.4 ker(T ) ={

[0 0 ]

} = {O} ( Example 5.2.7(2) ). Dimension Theorem, dim(T (R²)) + dim(ker(T )) = 2 + 0 = 2.

T₁, T₂ Rⁿ R^m linear transformation , c₁, c₂∈ R,

Rⁿ R^m linear transformation c1T1+ c2T2 ( Proposition 5.1.6).

c₁T₁+ c₂T₂ standard matrix representation T₁, T₂ standard matrix representation . , T R^m R^k linear transformation,

數 T◦ T1 Rⁿ R^k linear transformation ( Proposition 5.1.7). ,

T◦ T1 standard matrix representation T₁, T standard matrix representation .

Lemma 5.3.8. T₁, T₂ Rⁿ R^m linear transformations, T R^m R^k linear transformation. [T₁], [T₂] [T ] T1, T₂ T standard matrix representation.

(1) c₁, c₂∈ R, c₁T₁+ c₂T₂:Rⁿ→ R^m standard matrix representation c1[T1] + c2[T2],

[c₁T1+ c₂T2] = c₁[T₁] + c₂[T₂].

(2) T◦ T1:Rⁿ→ R^k standard matrix representation [T ][T₁], [T◦ T1] = [T ][T₁].

Proof. (1) v∈ Rⁿ, (c₁T1+ c₂T2)(v) = c₁T1(v) + c₂T2(v). standard matrix representation T₁(v) = [T₁]v, T₂(v) = [T₂]v,

(c1T1+ c2T2)(v) = c1[T1]v + c2[T2]v = (c1[T1] + c2[T2])v.

言 , c1[T1] + c2[T2] m×n matrix c1T1+ c2T2:Rⁿ→ R^m standard matrix representation , standard matrix representation 性 (Theorem 5.3.1) [c₁T1+ c₂T2] = c₁[T₁] + c₂[T₂].

(2) v∈ Rⁿ, (T◦ T1)(v) = T (T₁(v)). standard matrix rep- resentation T1(v) = [T1]v, (T◦ T1)(v) = T ([T1]v). , w∈ R^m

T (w) = [T ]w, (T ◦ T1)(v) = T ([T₁]v) = [T ]([T₁]v).

[T ]([T₁]v) = ([T ][T₁])v. 言 , [T ][T1] k× n matrix T◦ T1 :Rⁿ→ R^k standard matrix representation (T◦T1)(v) = ([T ][T₁])v, standard matrix repre-

sentation 性 (Theorem 5.3.1) [T◦ T1] = [T ][T₁]. 

Example 5.3.9. Example 5.3.3 linear transformations standard matrix

representations 數.

5.3. Matrix Representation 105

T :R³→ R² T (



 x₁ x2

x3



) =[

x₁+ x₂ x1− x3

]

. T standard matrix repre-

sentation [T ] =

[ 1 1 0 1 0 −1

]

. T^′:R²→ R³ T^′( [ x₁

x2

] ) =



 x₁ x1+ x₂ x1− x2



.

T^′ standard matrix representation [T^′] =



 1 0

1 1

1 −1



. 數

T^′◦ T : R³→ R³ (T^′◦ T)(



 x1

x2

x3



) = T^′(

[ x1+ x₂ x1− x3

] ) =



 x1+ x₂ (x1+ x2) + (x1− x3) (x1+ x2)− (x1− x3)



 =



 x1+ x₂ 2x1+ x2− x3

x2+ x3



.

, T^′◦ T standard matrix representation [T^′◦ T] =



 1 1 0 2 1 −1

0 1 1



.

, , [T^′][T ] =



 1 0 1 1 1 −1



[

1 1 0

1 0 −1 ]

=



 1 1 0 2 1 −1

0 1 1



.

[T^′◦ T] = [T^′][T ].

matrix linear transformation. , linear

transformation matrix 性 . T :Rⁿ→ R^m, T^′:R^m→ R^k linear transformations. T range R^m subspace, T (Rⁿ)⊆ R^m, (T^′◦T)(Rⁿ) = T^′(T (Rⁿ))⊆ T^′(R^m). T^′◦ T : Rⁿ→ R^k linear transformation range

T^′ range subspace. subspace dimension (Corollary 4.3.6),

dim((T^′◦ T)(Rⁿ))≤ dim(T^′(R^m)). 性 .

Proposition 5.3.10. A m×n matrix, B k×m matrix, rank(BA)≤ rank(A) rank(BA)≤ rank(B).

Proof. T :Rⁿ→ R^m T (v) = Av,∀v ∈ Rⁿ, T^′:R^m→ R^k T^′(w) = Bw,

∀w ∈ R^m. , [T ] = A, [T^′] = B Lemma 5.3.8 [T^′◦ T] = [T^′][T ] = BA.

前 dim((T^′◦ T)(Rⁿ))≤ dim(T^′(R^m)), Corollary 5.3.5 dim((T^′◦ T)(Rⁿ)) = rank([T^′◦ T]) = rank(BA) dim(T^′(R^m)) = rank([T^′]) = rank(B), rank(BA)≤ rank(B).

, , rank(A^TB^T)≤ rank(A^T). rank(A^T) = rank(A) rank((BA)^T) = rank(BA) (Proposition 4.4.14),

rank(BA) = rank((BA)^T) = rank(A^TB^T)≤ rank(A^T) = rank(A).



, 數 invertible ( one-to-one onto) , inverse

( 數) . invertible linear transformation, standard matrix