S is a basis for V

 Corollary 1.10.1: Every basis for V has the same number of vectors.

 dimension: the number of vectors in a basis

 Corollary 1.10.2: Let dim(V )=n.

1. S generates V. ⇒ |S| ≥ n, S can be reduced to a basis.

2. S generates V and |S| = n. ⇒ S is a basis for V . 3. S is LI. ⇒ |S| ≤ n, S can be extended to a basis.

 Theorem 1.11: dim(V ) < ∞; W is a subspace of V . Then 1. dim(W ) ≤ dim(V )

2. dim(W )=dim(V ) ⇒ W = V .

 Corollary 1.11: dim(V ) < ∞; W is a subspace of V . Then any basis for W can be extended to a basis for V .

Chapter 2 Linear transformation and Matrices

 linear transformation T : V → W for vector spaces V (F ) and W (F ) :, T (ax + by) = aT (x) + bT (y).

 function f : X → Y : ∀x ∈ X, ∃ unique f (x) ∈ Y

 domain of f : X

 codomain of f : Y

 range of f : f (X) = {f (x) : x ∈ X}

 image of A under f : f (A) = {f (x) : x ∈ A}

 preimage of B under f : f⁻¹(B) = {x : f (x) ∈ B};

also called as inverse image

 onto: f (X) = Y

 one-to-one: f (u) = f (v) ⇒ u = v

 inverse of f : f⁻¹ : Y → X such that

∀x ∈ X, f⁻¹(f (x)) = x; and ∀y ∈ Y, f (f⁻¹(y)) = y

 invertible f : f⁻¹ exists (⇔ one-to-one and onto)

 restriction of f to A: f_A : A → Y such that

∀x ∈ A, f_A = f (x)

 composite or composition of f : f : X → Y and g : Y → Z:

g ◦ f : X → Z such that ∀x ∈ X, (g ◦ f )(x) = g(f (x)) (We will use the notation gf in place of g ◦ f .)

 examples

 T_θ : R² → R², rotation by θ:

T_θ((a₁, a₂)) = (a₁ cos θ − a₂ sin θ, a₁ sin θ + a₂ cos θ)

 T : R² → R², projection on the y-axis: T ((a₁, a₂)) = (0, a₂) linearity: T (c(a₁, a₂) + d(b₁, b₂)) = T (ca₁ + db₁, ca₂ + db₂) = (0, ca₂ + db₂) = (0, ca₂) + (0, db₂) = c(0, a₂) + d(0, b₂) =

cT ((a₁, a₂)) + dT ((b₁, b₂))

 V = C(R, R): space of continuous functions from R to R T : V → R, integration over [a, b] : T (f ) = R _b

a f (t) dt linearity: T (cf + dg) = R _b

a(cf (t) + dg(t)) dt = c R _b

a f (t) dt + d R _b

a g(t) dt = cT (f ) + dT (g)

 I_V : V → V , identity transformation: I_V (v) = v

 T₀ : V → W , zero transformation: T₀(v) = 0

 null space and range

 null space of T : T : V → W : N (T ) = {x ∈ V : T (x) = 0}

 range space of T : T : V → W : R(T ) = {T (x) : x ∈ V }

 The null space and range of a linear transformation cannot be empty because T (0) = 0. That is, 0 ∈ N (T ) and 0 ∈ R(T )

 Another name for the null space is kernel.

Another name for the range is image.

 example:

 T : R² → R², projection on y-axis: T ((a₁, a₂)) = (0, a₂) N (T ) = {(a₁, 0) : a₁ ∈ R}, R(T ) = {(0, a2) : a₂ ∈ R}

 V = C(R, R): space of continuous functions T : V → R, integration over [a, b]: T (f ) = R _b

a f (t) dt N (T ) =

n

f : R _b

a f (t) dt = 0 o

, R(T ) = R

 Theorem 2.1: V and W are vector spaces; T : V → W is a linear transformation. Then N (T ) and R(T ) are subspaces of V and W , respectively.

proof for N (T ): x, y ∈ N (T ), a, b ∈ F

⇒ T (x) = T (y) = 0

⇒ T (ax + by) = aT (x) + bT (y) = 0 [linearity]

⇒ ax + by ∈ N (T )

 Theorem 2.2: T : V → W is a linear transformation; β = {v₁, · · · , v_n} is a basis for V . Then, R(T ) = span(T (β)) = span({T (v₁), · · · , T (v_n)}).

proof:

(i) T (β) ⊆ R(T ) [range]

⇒ span(T (β)) ⊆ R(T ) [R(T ) is a subspace; Theorem 1.5]

(ii) w ∈ R(T ) ⇒ ∃v such that T (v) = w [range]

⇒ v = Σa_iv_i [basis]

⇒ w = T (v) = T (Σa_iv_i) = Σa_iT (v_i) ∈ span(T (β)))

⇒ R(T ) ⊆ span(T (β))

(i), (ii) ⇒ R(T ) = span(T (β))

 Is {T (v₁), · · · , T (v_n)} a basis for R(T )?

 nullity and rank

 nullity of T : nullity(T ) = dim(N (T ))

 rank of T : rank(T ) = dim(R(T ))

 Theorem 2.3 (dimension theorem): T : V → W is a linear trans- formation; V < ∞. Then, nullity(T ) + rank(T ) = dim(V ).

proof: Assume the following:

 dim(V ) = n, nullity(T ) = k

 {v₁, · · · , v_k} is a basis for N (T ), and it extends to a basis {v₁, · · · , v_k, v_k+1, · · · , v_n} for V .

 S = {T (v_k+1), · · · , T (v_n)}

We will then show that S is a basis for R(T ).

“S generates R(T )”: Let w ∈ R(T ).

⇒ ∃v ∈ V such that T (v) = w. [range]

⇒ v = P_n

i=1 a_iv_i. [basis for V ] T (v) = w = P_n

i=1 a_iT (v_i) = P_n

i=k+1 a_iT (v_i) [null space]

w ∈ span(S)

“S is linearly independent”: Let P_n

i=k+1 b_iT (v_i) = 0.

⇒ T (P_n

i=k+1 b_iv_i) = 0 [linearity]

⇒ P_n

i=k+1 b_iv_i ∈ N (T ) [null space]

⇒ P_n

i=k+1 b_iv_i = P_k

i=1 c_iv_i for some c_i’s [basis for N (T )]

⇒ P_k

i=1 c_iv_i + P_n

i=k+1(−b_i)v_i = 0

⇒ c₁ = · · · = c_k = b_k+1 = · · · = b_n = 0 [basis for V ]

 This dimension theorem holds only for finite dimensional vec- tor spaces.

 example:

 T_θ: R² → R², rotation by θ:

T_θ((a₁, a₂)) = (a₁ cos θ − a₂ sin θ, a₁ sin θ + a₂ cos θ) dim(V ) = 2 = 2 + 0 = rank(T ) + nullity(T )

 T_θ: R² → R², projection on the y-axis: T_θ((a₁, a₂)) = (0, a₂) dim(V ) = 2 = 1 + 1 = rank(T ) + nullity(T )

 Theorem 2.4: T : V → W is a linear transformation. Then the following holds.

T is one-to-one ⇔ N (T ) = {0}.

proof: “⇒”: Assume T is one-to-one and v ∈ N (T ).

⇒ T (u + v) = T (u) + T (v) = T (u) [null space]

⇒ u + v = u [one-to-one]

⇒ v = 0 [cancellation law]

“⇐”: Assume N (T ) = {0} and T (u) = T (v).

⇒ T (u) − T (v) = T (u − v) = 0 [linear]

⇒ u − v ∈ N (T ) ⇒ u − v = 0 [assumption] ⇒ u = v

 Theorem 2.5: T : V → W is a linear transformation; dim(V ) = dim(W ) < ∞.

Then the followings are equivalent.

1. T is one-to-one.

2. T is onto.

3. rank(T ) = dim(V )

? The following figure serves as an intuitive proof.

 Rotation by θ is an example for the last two theorems.

 example: T : P₂(R) → P3(R), T (f )(x) = 2f (x) + ₀^x 3f (t) dt

? by theorem 2.2,

R(T ) = span(T (1), T (x), T (x²) ) = span(



3x, 2 + 3

2x², 4x + x³

 )



3x, 2 + 3

2x², 4x + x³



is linearly independent.

⇒ It is a basis for R(T ) ⇒ rank(T ) = 3

⇒ nullity(T ) = 0 [dim thm] ⇒ T is one-to-one. [Thm 2.4]

But T is not onto. [dim(V ) = 3 6= dim(W ) = 4]

 example: T : P₂(R) → P2(R), T (f )(x) = 2f⁰(x) − 3f (x)

? by theorem 2.2,

R(T ) = span(T (1), T (x), T (x²) ) = span(−3, 2 − 3x, 4x − 3x² )

⇒ rank(T ) = 3 = dim(P₂(R))

⇒ T is onto; T is one-to-one. [Theorem 2.5]

 Theorem 2.6: {v₁, · · · , v_n} is a basis for V ; w₁, · · · , w_n ∈ W . Then there exists only one linear transformation T : V → W such that T (v_i) = w_i, i = 1, · · · , n.

proof: Let x = Σa_iv_i, and define T as follows:

∀x ∈ V , T (x) = T (Σa_iv_i) = Σa_iw_i. [linear? unique?]

“linearity”: T (cx + dy) = T (cΣa_iv_i + dΣb_iv_i) [y = Σb_iv_i]

= T (Σ(ca_i + db_i)v_i) = Σ(ca_i + db_i)w_i [def of T ]

= cΣa_iw_i + dΣb_iw_i = cT (Σa_iv_i) + dT (Σb_iv_i) [def of T ]

= cT (x) + dT (y)

“uniqueness”: Let a linear transformation U satisfy U (v_i) = w_i.

∀x ∈ V , U (x) = U (Σa_iv_i) = Σa_iU (v_i) = Σa_iw_i = T (x)

⇒ U = T

dependent or even repeated.

 This theorem says that in order to specify a linear transforma- tion, you only need to specify it on a basis.

 It also implies that R(T ) = span({w₁, · · · , w_n}) by theorem 2.2, whereby you can design R(T ) as you wish.

 This theorem is analogous to:

 that for two real numbers v and w, there is only one linear function f : R → R such that f (v) = w. [straight line]

 that for six real numbers; v₁₁, v₁₂, v₂₁, v₂₂, w₁, w₂, there is only one linear function f R² → R such that f((v11, v₁₂)) = w₁ and f ((v₂₁, v₂₂)) = w₂. [plane]

⌅⌅ Null space and Range space

⌅ null space of T : T : V ! W : N(T ) = {x 2 V : T (x) = 0}

⌅ range space of T : T : V ! W : R(T ) = {T (x) : x 2 V }

⌅⌅ Theorem 2.1: V and W are vector spaces; T : V ! W is a linear transformation. Then N(T ) and R(T ) are subspaces of V and W , respectively.

⌅⌅ Theorem 2.2: T : V ! W is a linear transformation; = {v1, · · · , vⁿ} is a basis for V . Then, R(T ) = span(T ( )) = span({T (v1), · · · , T (vⁿ)}).

⌅ {T (v1), · · · , T (vⁿ)} is not a basis for R(T )?

V쌔0-0t2-R^(T)

⌅⌅ nullity and rank

⌅ nullity of T : nullity(T ) = dim(N(T ))

⌅ rank of T : rank(T ) = dim(R(T ))

⌅⌅ Theorem 2.3 (dimension theorem): T : V ! W is a linear trans- formation; V < 1. Then, nullity(T ) + rank(T ) = dim(V ).

⌅ This dimension theorem holds only for finite dimensional vector spaces.

⌅⌅ Theorem 2.4: T : V ! W is a linear transformation. Then the following holds.

T is one-to-one , N(T ) = {0}.

⌅⌅ Theorem 2.5: T : V ! W is a linear transformation; dim(V ) = dim(W ) < 1. Then the followings are equivalent.

1. T is one-to-one.

2. T is onto.

3. rank(T ) = dim(V )

Et

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⌅⌅ Theorem 2.6: {v1, · · · , vn} is a basis for V ; w1, · · · , wn 2 W . Then there exists only one linear transformation T : V ! W such that T (v_i) = w_i, i = 1, · · · , n.

proof: Let x = ⌃a_iv_i, and define T as follows:

8x 2 V , T (x) = T (⌃aiv_i) = ⌃a_iw_i. [linear? unique?]

“T (v_i) = w_i”: clear by letting a_i = 1, a_j = 0) for j 6= i

“linearity”: T (⌃a_iv_i) = ⌃a_iT (v_i), 8{a1, ..., a_n}

“uniqueness”: Let a linear transformation U satisfy U(v_i) = w_i. 8x 2 V , U(x) = U(⌃aiv_i) = ⌃a_iU (v_i) = ⌃a_iw_i = T (x)

) U = T

E, Estate

몸

⌅ Note that w_i’s can be arbitrary vectors in W , possibly linearly dependent or even repeated.

⌅ This theorem says that in order to specify a linear transforma- tion, you only need to specify it on a basis.

⌅ It also implies that R(T ) = span({w1, · · · , wⁿ}) by theorem 2.2, whereby you can design R(T ) as you wish.

⌅ This theorem is analogous to:

⌅⌅ that for two real numbers v and w, there is only one linear function f: R ! R such that f(v) = w. [straight line]

⌅⌅ that for six real numbers; v₁₁, v₁₂, v₂₁, v₂₂, w₁, w₂, there is only one linear function f R² ! R such that f((v11, v₁₂)) = w₁ and f((v₂₁, v₂₂)) = w₂. [plane]

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Matrix representation of a linear transformation

⌅⌅ Recall that given a basis = {v1, · · · , vⁿ} for V , we can write x = ⌃c_iv_i for any x 2 V and that [v] = (c1, · · · , cⁿ)^t is unique by theorem 1.8 and is called the (n-tuple) representation of x in or relative to .

⌅ [v] = (0, · · · , 0, 1, 0, · · · , 0)^t, in which 1 is the i-th element.

⌅⌅ ordered basis: basis with a given order of the elements

⌅ If the basis is considered as a set, i.e. not ordered, the represen- tation is unique only up to a permutation.

⌅ From now on all the bases are assumed ordered.

⌅⌅ example: = {e1, · · · , eⁿ}, standard basis for Fⁿ, where

e₁ = (1, 0, · · · , 0)^t, e₂ = (0, 1, · · · , 0)^t,· · · ,en = (0, 0, · · · , 1)^t n = 3 ) [(2, 3, 1)] = 2, 3, 1^t

I

i

_

=

1 ) _{, If} Bit와의 ^.es

}

⌅⌅ example: V = P₂(R), f(x) = 4 + 6x 7x²

= 1, x, x² , standard basis, [f] = (4, 6, 7)^t

= 1, 1 + x, 1 + x + x² [f ] = (c₁, c₂, c₃)^t

) f = c1 + c₂(1 + x) + c₃(1 + x + x²)

) 4 + 6x 7x² = (c₁ + c₂ + c₃) + (c₂ + c₃)x + c₃x² ) [f] = ( 2, 13, 7)^t

⌅⌅ Consider T : V ! W , a basis for V , and a basis for W . Then what is the relationship between [x] and [T (x)] ?

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Page 8

⌅⌅ matrix multiplication rule:

A: m⇥n; B: n⇥p; C = AB ) C: m⇥p; Cij = P_m

k=1 A_ikB_kj

⌅ example: a b ·

✓c d

◆

= ac + bd,

✓a b

◆

· c d =

✓ac ad bc bd

◆

✓a b c d e f

◆

· 0

@g h i j k l

1 A=

✓ag + bi + ck ah + bj + cl dg + ei + f k dh + ej + f l

◆

⌅⌅ matrix representation of a linear transform:

= {v1, · · · , vⁿ} is a basis for V ;

= {w1, · · · , wⁿ} is a basis for W ; and T : V ! W is a linear transformation.

s^{i.tt ww}

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GH

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T의 Matrix

Rap

⇒

) 8x 2 V , x = P_n

j=1 c_jv_j

) [x] = (c1, · · · , cⁿ)^t representation of x in ) T (x) = T (P_n

j=1 c_jv_j) = P_n

j=1 c_jT (v_j) T (v_j) 2 W ) T (vj) = P_m

i=1 a_ijw_i for some a_ij ) T (x) = P_n

j=1 c_jT (v_j) = P_n

j=1 c_j P_m

i=1 a_ijw_i ) T(x) = c₁(a₁₁w₁ + a₂₁w₂ + · · · + am1w_m)

+ ...

= + c_n(a_1nw₁ + a_2nw₂ + · · · + a^mnw_m) ) T (x) = P_m

i=1(P_n

j=1 a_ijc_j)w_i ) [T (x)] =

P_n

j=1a_1jc_j P_n ...

j=1a_mjc_j

!

=

a₁₁ · · · a1n

... ...

a_m1 · · · amn

!

·

c₁ ...

c_n

!

= A[x]

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A = [T ] is the matrix representation of T in and . ) T (x) = [T ] [x]

[T ] = ([T (v₁)] , · · · , [T (vⁿ)] ) =

a₁₁ ...

a_m1

!

· · ·

a_1n ...

a_mn

!!

= A

⌅ The matrix representation of T is unique by Theorem 1.8.

⌅ T (x) = y ) [T ] [x] = y

⌅ You can remember this notation by relating it to · = .

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⌅ If V = W and = , the notation [T ] simplifies to [T ] . ) [T (x)] = [T ] [x]

That is, [T ] is the matrix representation of T in and .

) [T ] = ([T (v1)] , · · · , [T (vⁿ)] ) = 0

@a₁₁ · · · a1n

... ...

a_n1 · · · aⁿⁿ 1 A, where = {v1, · · · , vⁿ}.

When the domain and the codomain are the same vector space, the linear transformation is called the _.linear operator.

⌅⌅ example: T : P₃(R) ! P2(R), T (f) = f⁰;

= 1, 1 + x, 1 + x + x², 1 + x + x² + x³ is a basis for P₃(R);

and = 1, x, x² is a basis for P₂(R).

T (1) = 0 = 0 · 1 + 0 · x + 0 · x² T (1 + x) = 1 = 1 · 1 + 0 · x + 0 · x² T (1 + x + x²) = 1 + 2x = 1 · 1 + 2 · x + 0 · x² T (1 + x + x² + x³) = 1 + 2x + 3x² = 1 · 1 + 2 · x + 3 · x² [T ] =

0

@0 1 1 1 0 0 2 2 0 0 0 3

1

A, [4 6x + 3x³] = (10, 6, 3, 3)^t

T (4 6x + 3x³) = 6 + 9x² ) 0

@0 1 1 1 0 0 2 2 0 0 0 3

1 A ·

0 BB

@ 10

6 3 3

1 CC A =

0

@ 6

0 9

1 A

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⌅⌅ addition and scalar multiplication of linear transformations:

⌅ Definition. Let T , U: V (F ) ! W (F ); a 2 F be arbitrary functions. We define

1. T + U: V ! W by (T + U)(x) = T (x) + U(x);

2. aT : V ! W by (aT )(x) = aT (x)

⌅⌅ Theorem 2.7: Let T , U: V (F ) ! W (F ); be linear transforma- tions. Then the followings are true.

1. 8a, b 2 F , aT + bU is a linear transformation.

2. The set of all linear transformations from V to W becomes a vector space over F .

⌅ T₀, the zero transformation, plays the role of the zero vector in the vector space.

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⌅⌅ space of linear transformations:

L(V, W ) : vector space of all linear transformations from V to W L(V ) : vector space of all linear transformations from V to V

⌅⌅ Theorem 2.8: T , U: V (F ) ! W (F ); are linear transformations;

dim(V ), dim(W ) < 1; a 2 F . Then the followings are true.

1. [T + U] = [T ] + [U]

2. [aT ] = a[T ]

⌅ By this theorem, given bases for V , for W , dim(V ) = n, and dim(W ) = m, the vector space L(V, W ) can be identified by the vector space of M_m⇥n.

⌅⌅ example: T ,U: P₂(R) ! P2(R), T (f) = f⁰, U(f) = f 2f⁰;

= 1, 1 + x, 1 + x + x² ; = 1, x, x² . (4T + 2U )(f ) = 4f⁰ + 2(f 2f⁰) = 2f

[T ] = 0

@0 1 1 0 0 2 0 0 0

1

A; [U] = 0

@1 1 1

0 1 3

0 0 1 1 A;

[4T + 2U ] = 0

@2 2 2 0 2 2 0 0 2

1

A = 4[T ] + 2[U]

[T ] = 0

@0 1 0 0 0 2 0 0 0

1

A; [U] = 0

@1 2 0

0 1 4

0 0 1

1

A; [4T + 2U] = 0

@2 0 0 0 2 0 0 0 2

1 A.

T111 ^{= 0}

(

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4

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T1조) ^{= 1} U ^1조) -27C

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Composition of linear transformations

⌅⌅ The composition UT of T : V ! W and U: W ! Z is a linear transformation such that 8x 2 V , (UT )(x) = U(T (x))

⌅ This composition is a linear transformation from V to Z.

⌅⌅ example: T ,U: P₂(R) ! P2(R), T (f) = f⁰, U(f) = f 2f⁰. U (T (f )) = T (f ) 2(T (f ))⁰ = f⁰ 2f⁰⁰

T (U (f )) = (U (f ))⁰ = f⁰ 2f⁰⁰

It happens that U(T (f)) = T (U(f)), but this is not always the case.

⌅⌅ example: T ,U: P₂(R) ! P2(R), T (f) = xf⁰, U(f) = f⁰. U (T (f )) = (T (f ))⁰ = (xf⁰)⁰ = f⁰ + xf⁰⁰

T (U (f )) = x(U (f ))⁰ = x(f⁰)⁰ = xf⁰⁰ ) U(T (f)) 6= T (U(f))

⌅⌅ Theorem 2.9 and 2.10: S, T , T₁, T₂, U, U₁, and U₂ are linear transformations with appropriate domains and co-domains; I is an identity transformation; a 2 F . Then we have as belows:

1. UT is a linear transform.

2. distributivity(1): U(T₁ + T₂) = U T₁ + U T₂ 3. distributivity(2): (U₁ + U₂)T = U₁T + U₂T 4. associativity: S(UT ) = (SU)T

5. identity: IT = T I = T (Note that the two I’s are different.) 6. a(UT ) = (aU)T = U(aT )

U 5127GBP ^EU ( ^{Na) t} p^Tag))

= 2 UTCX⁾

tpngjsee.tt#bydefiniti@(UtT)bc)=Ub4tTbc

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⌅⌅ matrix representation of composition

For j = 1, · · · , n,

(U T )(v_j) = U (T (v_j)) = U (P_m

k=1 B_kjw_k) = P_m

k=1 B_kjU (w_k)

= P_m

k=1 B_kj(P_p

i=1 A_ikz_i) = P_p

i=1(P_m

k=1 A_ikB_kj)z_i

= Pp

i=1 C_ijz_i ! j^th column of C.

⌅⌅ Theorem 2.11: [UT ]_↵ = C = AB = [U ] [T ]_↵

⌅ This equation is related to

↵ = · ↵.

⌅⌅ example:

U: P₃(R) ! P2(R), U(f) = f⁰ T : P₂(R) ! P3(R), T (f) = R _x

0 f (t) dt

[U T ]_↵ = [U ]^↵[T ]_↵ = 0

@0 1 0 0 0 0 2 0 0 0 0 3

1 A ·

0 BB

@

0 0 0 1 0 0 0 ¹₂ 0 0 0 ¹₃

1 CC A =

0

@1 0 0 0 1 0 0 0 1

1 A =

[I_P2]_↵

[T U ] = [T ]_↵[U ]^↵ = 0 BB

@

0 0 0 1 0 0 0 ¹₂ 0 0 0 ¹₃

1 CC A ·

0

@0 1 0 0 0 0 2 0 0 0 0 3

1 A =

0 BB

@

0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

1 CC A 6=

[I_P3]

⌅⌅ identity matrix: I: I_ij = _ij =

(1, if i = j

0, else (i 6= j), where _ij is the Kronecker Delta.

⌅⌅ Theorem 2.12: A, A₁, A₂, m⇥n; B, B1, B₂, ,n⇥p; and C, p⇥

q, are matrices; I_m and I_n are identity matrices of the respective sizes; a 2 F . Then we have the followings.

1. AB is a matrix of the size m ⇥ p.

2. distributivity(1): A(B₁ + B₂) = AB₁ + AB₂ 3. distributivity(2): (A₁ + A₂)B = A₁B + A₂B 4. associativity: (AB)C = A(BC)

5. identity: I_mA = AI_n = A 6. a(AB) = (aA)B = A(aB)