Euclidean n & Vector Spaces. Matrices & Vector Spaces

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Matrices & Vector Spaces

Euclidean –n & Vector Spaces

Lecture 9

Delivered by:

Filson Maratur Sidjabat [email protected]

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(90%*score / 20% extra points for HW-Q)

Retake Quiz

1. Compute (a) det(a), (b) adj(A), and (c) A^-1 of this matrix:

 A = 1 3 1

2 1 1

−2 2 −1

2. Use Cramer’s rule to solve

 𝑥₁ + 2𝑥₂ + 𝑥₃ = 5

 2𝑥₁ + 2𝑥₂ + 𝑥₃ = 6

 𝑥₁ + 2𝑥₂ + 3𝑥₃ = 9

2015/6/4 Elementary Linear Algebra 2

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Use adjoint for this problems

1. Find an elementary matrix E such that EB = D (20 mark) 2. Find an elementary matrix F such that AF = C (20 mark)

𝐴 = 2 1 3 4 2 7

1 3 5 𝐵 = 3 1

5 2 𝐶 = 0 1 3 0 2 7

−3 3 5 𝐷 = 1 2 3 4

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Preview

 Sistem Persamaan Linier

 Eliminasi Gauss dan Gauss-Jordan

 Matriks dan Operasi Matriks

 Invers Matriks

 Invers dan Aritmetika Matriks

 Matriks Elementer dan Metode mencari A^-1

 Determinan

 Cofactor Expansion

 Adjoint and Cramer’s Rule

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Matrices & Vector Spaces Vector Spaces

Lecture 9 (Make-up class)

Delivered by:

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Vector - quick reminder

 Jika diketahui: v = (2,0, -5) dan w = (3,1,4), maka:

 v+w

 3v

 -w

 v-w

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Vector - quick reminder

 v . w = ||v||.||w|| cos q (hasil kali titik - proyeksi)

 v x w (hasil kali silang)

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Geometry of Vectors

 Vectors have direction and magnitude

 The are portable

 They are added (subtracted) tip-to-tail

 Parallelogram rule applies

 Three-element vector is three dimensional space

 More than three elements is called n-tuple

 Has no geometric representation but still used extensively

 Good idea to draw vectors

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Matrices & Vector Spaces Euclidean n-Space

Lecture 9

#4^th June 2015 Delivered by:

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4-1 Definitions

 If n is a positive integer, then an ordered n-tuple is a sequence of n real numbers (a₁,a₂,…,a_n). The set of all ordered n-tuple is called n- space and is denoted by Rⁿ.

 Two vectors u = (u₁,u₂,…,u_n) and v = (v₁,v₂,…, v_n) in Rⁿ are called equal if

u₁ = v₁,u₂ = v₂, …, u_n = v_n The sum u + v is defined by

u + v = (u₁+v₁, u₁+v₁, …, u_n+v_n)

and if k is any scalar, the scalar multiple ku is defined by ku = (ku₁,ku₂,…,ku_n)

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4-1 Remarks

 The operations of addition and scalar multiplication in this definition are called the standard operations on Rⁿ.

 The zero vector in Rⁿ is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0).

 If u = (u₁,u₂,…,u_n) is any vector in Rⁿ, then the negative (or additive inverse) of u is denoted by -u and is defined by -u = (-u₁,-u₂,…,-u_n).

 The difference of vectors in Rⁿ is defined by

v – u = v + (-u) = (v₁– u₁,v₂– u₂,…,v_n– u_n)

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Theorem 4.1.1 (Properties of Vector in R

ⁿ

)

 If u = (u₁,u₂,…,u_n), v = (v₁,v₂,…, v_n), and w = (w₁,w₂,…, w_n) are vectors in Rⁿ and k and l are scalars, then:

 u + v = v + u

 u + (v + w) = (u + v) + w

 u + 0 = 0 + u = u

 u + (-u) = 0; that is u – u = 0

 k(lu) = (kl)u

 k(u + v) = ku + kv

 (k+l)u = ku+lu

 1u = u

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4-1 Euclidean Inner Product

 Definition

 If u = (u₁,u₂,…,u_n), v = (v₁,v₂,…, v_n) are vectors in Rⁿ, then the Euclidean inner product u · v is defined by

u · v = u₁v₁+ u₂v₂+… + u_nv_n

 Example 1

 The Euclidean inner product of the vectors u = (-1,3,5,7) and v = (5,-4,7,0) in R⁴ is

u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

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Theorem 4.1.2

Properties of Euclidean Inner Product

 If u, v and w are vectors in Rⁿ and k is any scalar, then

 u · v = v · u

 (u + v) · w = u · w + v · w

 (k u) · v = k(u · v)

 v · v ≥ 0; Further, v · v = 0 if and only if v = 0

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4-1 Example 2

 (3u + 2v) · (4u + v)

= (3u) · (4u + v) + (2v) · (4u + v )

= (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v

=12(u · u) + 11(u · v) + 2(v · v)

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4-1 Norm and Distance in Euclidean n-Space

 We define the Euclidean norm (or Euclidean length) of a vector u = (u₁,u₂,…,u_n) in Rⁿ by

 Similarly, the Euclidean distance between the points u = (u₁,u₂,…,u_n) and v = (v₁, v₂,…,v_n) in Rⁿ is defined by

2 2

1 2

/

1 ...

)

(   u u  u_n

 u u u

2 2

2 1

1 ) ( ) ... ( )

( )

,

( u v u v u_n v_n

d u v  u v       

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4-1 Example 3

 Example

 If u = (1,3,-2,7) and v = (0,7,2,2), then in the Euclidean space R⁴

58 )

2 7 ( )

2 2 ( ) 7 3 ( )

0 1 ( )

, (

7 3 63 )

7 ( ) 2 ( )

3 ( )

1 (

2 2

























 v u u d

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Theorem 4.1.3

(Cauchy-Schwarz Inequality in R ⁿ )

 If u = (u₁,u₂,…,u_n) and v = (v₁, v₂,…,v_n) are vectors in Rⁿ, then |u · v| ≤ || u || || v ||

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Theorem 4.1.4

(Properties of Length in R ⁿ )

 If u and v are vectors in Rⁿ and k is any scalar, then

 || u || ≥ 0

 || u || = 0 if and only if u = 0

 || ku || = | k ||| u ||

 || u + v || ≤ || u || + || v || (Triangle inequality)

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Theorem 4.1.5

(Properties of Distance in R ⁿ )

 If u, v, and w are vectors in Rⁿ and k is any scalar, then

 d(u, v) ≥ 0

 d(u, v) = 0 if and only if u = v

 d(u, v) = d(v, u)

 d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality)

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Hasil kali Titik dari Vektor

Jika u dan v adalah vektor - vektor dalam ruang berdimensi 2 atau berdimensi 3 dan q adalah

sudut antara u dan v, maka hasil kali titik atau hasil kali dalam euclidean u.v, didefinisikan

sebagai :

 







 q

0 atau v

0 u

jika

0 0 dan v

0 u

jika

cos v

v u

.

u

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 u.v = u₁.v₁+ u₂.v₂+u₃.v₃ R₃

 u.v = u₁.v₁+ u₂.v₂ R₂

 CONTOH : u = (2,-1,1) DAN v = (1,1,2), CARILAH u.v dan tentukan sudut antara u dan v

v u

.

cos  q .

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Sudut Antar Vektor



Jika u dan v adalah vektor-vektor tak nol, maka :

v u

v

.

cos  q u

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Hasil kali titik bisa digunakan untuk memperoleh informasi mengenai sudut antara 2 vektor.



Jika u dan v adalah vektor-vektor tak nol dan q adalah sudut antara kedua vektor tersebut, maka :

q lancip jika dan hanya jika u.v>0 q tumpul jika dan hanya jika u.v<0

q =/2 jika dan hanya jika u.v=0

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

u.v = u

₁

.v

₁

+ u

₂

.v

₂

+u

₃

.v

₃

 R

₃



u.v = u

₁

.v

₁

+ u

₂

.v

₂

 R

₂

CONTOH :

u = (2,-1,1) dan v = (1,1,2),

Carilah u.v serta tentukan sudut antara u dan v

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4-1 Orthogonality

 Two vectors u and v in Rⁿ are called orthogonal if u · v = 0

 Example 4

 In the Euclidean space R⁴ the vectors

u = (-2, 3, 1, 4) and v = (1, 2, 0, -1) are orthogonal, since

u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0

 If u and v are called orthogonal, we writes: u  v

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Hasil Kali Silang Vektor

 Jika hasil kali titik berupa suatu skalar maka hasil kali silang berupa suatu vektor.

 Jika u = (u₁,u₂,u₃) dan v = (v₁,v₂,v₃) adalah vektor-vektor dalam ruang berdimensi 3, maka hasil kali silang u x v adalah vektor yang didefinisikan sebagai

u x v =(u₂v₃ - u₃v₂ ,u₃v₁ - u₁v₃ ,u1v₂ - u₂v₁ ) atau dalam notasi determinan :

 



 

 

 v v

u u

, v

v

u u

, v

v

u u x v u

2 1

3 1

3 2

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Sifat-sifat utama dari hasil kali silang.

 Jika u,v, dan w adalah sebarang vektor dalam ruang berdimensi 3 dan k adalah sebarang skalar, maka :

u x v = -(v x u)

u x (v+w) = (u x v) + (u x w) (u + v) x w = (u x w) + (v x w) k(u x v) = (ku) x v = u x (kv) u x 0 = 0 x u = 0

u x u = 0

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Hubungan antara hasil kali titik dan hasil kali silang

 Jika u, v dan w adalah vektor-vektor dalam ruang berdimensi 3, maka :

u.(u x v) = 0 u x v ortogonal terhadap u.

v.(u x v) = 0 u x v ortogonal terhadap v.

|u x v|²=|u|²|v|² – (u.v)² identitas Lagrange |u x v|=|u||v|sin Ө

u x (v x w) = (u.w)v – (u.v)w (u x v) x w = (u.w)v – (v.w)u

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Vector - quick reminder

 v . w = ||v||.||w|| cos q (hasil kali titik - proyeksi)

 v x w (hasil kali silang)

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Theorem 4.1.7

(Pythagorean Theorem in R ⁿ )

 If u and v are orthogonal vectors in Rⁿ which the

Euclidean inner product, then || u + v ||² = || u ||² + || v ||²

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4-1 Matrix Formulae for the Dot Product

 If we use column matrix notation for the vectors u = [u₁ u₂ … u_n]^T and v = [v₁ v₂ … v_n]^T , or

then

u · v = v^Tu Au · v = u · A^Tv u · Av = A^Tu · v





































n

n v

v u

u



1 1

and v u

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4-1 Example 5

 Verifying that Au‧v= u‧A^tv

























































5 0 2 ,

4 2 1 ,

1 0

1

1 4

2

3 2 1

v u

A

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4-1 A Dot Product View of Matrix Multiplication

 If A = [a_ij] is an mr matrix and B =[b_ij] is an rn matrix, then the ij- the entry of AB is

a_i1b_1j+ a_i2b_2j + a_i3b_3j + … + a_irb_rj

which is the dot product of the ith row vector of A and the jth column vector of B

 Thus, if the row vectors of A are r₁, r₂, …, r_m and the column vectors of B are c₁, c₂, …, c_n,

















2 2

2 1

2

1 2

1 1

1

n n

AB

c r

c r c

r c

r

c r c

r c

r







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4-1 Example 6

 A linear system written in dot product form system dot product form

0 8

5

5 4

7 2

1

4 3

3 2

1

3 2

1

3 2

1

















x x

x

x x

x

x x

x





































0 5 1

) ,

, ( (1,5,-8)

) ,

, ( (2,-7,-4)

) ,

, ( (3,-4,1)

3 2 1

x x x

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Homework

1. Gunakan vektor-vektor untuk mencari cosinus sudut dibagian dalam sudut segitiga dengan titik-titik sudut (-1, 0), (-2, 1) dan (1, 4)

2. Diketahui vektor u = ( 2, -3, 4 ) dan v = ( -1, 3, 2 ).

Berapakah nilai u x v ?

3. Carilah luas segitiga yang ditentukan oleh titik-titik A ( 2, 2, 0 ), B ( -1, 0, 2 ), C ( 0, 4, 3 ).

4. Misalkan u =(-1, 3, 2) w=(1, 1, -1). Cari semua vektor y yang memenuhi u x y = w!

Euclidean n & Vector Spaces. Matrices & Vector Spaces

Matrices & Vector Spaces