Euclidean n & Vector Spaces. Matrices & Vector Spaces

Lecture 9

Euclidean –n & Vector Spaces Delivered by: Filson Maratur Sidjabat [email protected]

Matrices & Vector Spaces #4th June 2015

(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: 

1 3 1 A= 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

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Elementary Linear Algebra

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Use adjoint for this problems 2 1 𝐴= 4 2 1 3

3 7 5

𝐵=

3 5

1 2

0 1 3 𝐶= 0 2 7 −3 3 5

𝐷=

1 2 3 4

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

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Preview 

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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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Lecture 9 (Make-up class)

Vector Spaces Delivered by: Filson Maratur Sidjabat [email protected]

Matrices & Vector Spaces #2 week of June 2014

Vector - quick reminder 

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

 

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v+w 3v -w v-w


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v . w = ||v||.||w|| cos q vxw

(hasil kali titik - proyeksi) (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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Lecture 9

Euclidean n-Space Delivered by: Filson Maratur Sidjabat [email protected]

Matrices & Vector Spaces #4th June 2015

4-1 Definitions 

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



Two vectors u = (u1 ,u2 ,…,un) and v = (v1 ,v2 ,…, vn) in Rn are called equal if u1 = v1 ,u2 = v2 , …, un = vn The sum u + v is defined by u + v = (u1+v1 , u1+v1 , …, un+vn) and if k is any scalar, the scalar multiple ku is defined by ku = (ku1 ,ku2 ,…,kun)

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





The difference of vectors in Rn is defined by v – u = v + (-u) = (v1 – u1 ,v2 – u2 ,…,vn – un)

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Theorem 4.1.1 (Properties of Vector in Rn) 

If u = (u1 ,u2 ,…,un), v = (v1 ,v2 ,…, vn), and w = (w1 ,w2 ,…, wn) are vectors in Rn and k and l are scalars, then: 

      

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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 ,…,un), v = (v1 ,v2 ,…, vn) are vectors in Rn, then the Euclidean inner product u · v is defined by u · v = u1 v1 + u2 v2 + … + un vn

Example 1 

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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 

 



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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 ,…,un) in Rn by u  (u  u)1/ 2  u12  u22  ...  un2



Similarly, the Euclidean distance between the points u = (u1 ,u2 ,…,un) and v = (v1 , v2 ,…,vn) in Rn is defined by d (u, v)  u  v  (u1  v1 ) 2  (u2  v2 ) 2  ...  (un  vn ) 2

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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 u  (1) 2  (3) 2  (2) 2  (7) 2  63  3 7 d (u, v )  (1  0) 2  (3  7) 2  (2  2) 2  (7  2) 2  58

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Theorem 4.1.3 (Cauchy-Schwarz Inequality in Rn) 

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

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Theorem 4.1.4 (Properties of Length in Rn) 

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

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|| 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 Rn) 

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

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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 :

 u v cos q jika u  0 dan v  0 u.v   jika u  0 atau v  0 0

 

u.v = u1.v1+ u2.v2+u3.v3  R3 u.v = u1.v1+ u2.v2  R2

u.v cos q  u.v 

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

Sudut Antar Vektor



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

u.v cos q  u v

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

 

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 serta tentukan sudut antara u dan v

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 :

 u2 u x v    v2

u3 v3

, 

u1 u 2   , v3 v1 v 2 

u1 u 3 v1

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) ux0=0xu=0 uxu=0

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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v . w = ||v||.||w|| cos q vxw

(hasil kali titik - proyeksi) (hasil kali silang)


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Theorem 4.1.7 (Pythagorean Theorem in Rn) 

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 … un]T and v = [v1 v2 … vn]T , or  u1  u     and un 

 v1  v     vn 

then

u · v = vTu Au · v = u · ATv u · Av = ATu · v 2015/6/4


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4-1 Example 5 

Verifying that Au‧v= u‧Atv  1  2 3  1   2 A   2 4 1, u   2 , v   0   1 0 1  4   5 

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4-1 A Dot Product View of Matrix Multiplication 



If A = [aij] is an mr matrix and B =[bij] is an rn matrix, then the ijthe entry of AB is ai1b1j + ai2b2j + ai3b3j + … + airbrj 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, …, rm and the column vectors of B are c1, c2, …, cn ,

r1  c1 r1  c 2  r1  c n r  c r  c  r  c 2 n AB   2 1 2 2      rm  c1 rm  c 2  rm  c n 2015/6/4


      35

4-1 Example 6 

A linear system written in dot product form system 3x1  4 x2  x3  1 2 x1  7 x2  4 x3  5 x1  5 x2  8 x3  0

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dot product form

 (3,-4,1)  ( x1 , x2 , x3 )  1 (2,-7,-4)  ( x , x , x )  5 1 2 3      (1,5,-8)  ( x1 , x2 , x3 )  0


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Homework 1.

2.

3.

4.

Gunakan vektor-vektor untuk mencari cosinus sudut dibagian dalam sudut segitiga dengan titik-titik sudut (-1, 0), (-2, 1) dan (1, 4) Diketahui vektor u = ( 2, -3, 4 ) dan v = ( -1, 3, 2 ). Berapakah nilai u x v ? Carilah luas segitiga yang ditentukan oleh titik-titik A ( 2, 2, 0 ), B ( -1, 0, 2 ), C ( 0, 4, 3 ). Misalkan u =(-1, 3, 2) w=(1, 1, -1). Cari semua vektor y yang memenuhi u x y = w!

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Euclidean n & Vector Spaces. Matrices & Vector Spaces

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