Euclidean n & Vector Spaces. Matrices & Vector Spaces

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1 Lecture 9 Euclidean n & Vector Spaces Delivered by: Filson Maratur Sidjabat fmsidjabat@president.ac.id Matrices & Vector Spaces #4 th June 05

2 (90%*score / 0% extra points for HW-Q) Retake Quiz. Compute (a) det(a), (b) adj(a), and (c) A - of this matrix: A = 3. Use Cramer s rule to solve x + x + x 3 = 5 x + x + x 3 = 6 x + x + 3x 3 = 9 05/6/4 Elementary Linear Algebra

3 Use adjoint for this problems A = B = 3 5 C = D = 3 4. Find an elementary matrix E such that EB = D (0 mark). Find an elementary matrix F such that AF = C (0 mark) 05/6/4 Elementary Linear Algebra 3

4 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 - Determinan Cofactor Expansion Adjoint and Cramer s Rule 05/6/4 Elementary Linear Algebra 4

5 Lecture 9 (Make-up class) Vector Spaces Delivered by: Filson Maratur Sidjabat fmsidjabat@president.ac.id Matrices & Vector Spaces # week of June 04

6 Vector - quick reminder Jika diketahui: v = (,0, -5) dan w = (3,,4), maka: v+w 3v -w v-w 05/6/4 Elementary Linear Algebra 6

7 Vector - quick reminder Jika diketahui: v = (,0, -5) dan w = (3,,4), maka: v. w = v. w cos q (hasil kali titik - proyeksi) v x w (hasil kali silang) 05/6/4 Elementary Linear Algebra 7

8 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 05/6/4 Elementary Linear Algebra 8

9 05/6/4 Elementary Linear Algebra 9

10 Lecture 9 Euclidean n-space Delivered by: Filson Maratur Sidjabat fmsidjabat@president.ac.id Matrices & Vector Spaces #4 th June 05

11 4- Definitions If n is a positive integer, then an ordered n-tuple is a sequence of n real numbers (a,a,,a n ). The set of all ordered n-tuple is called n- space and is denoted by R n. Two vectors u = (u,u,,u n ) and v = (v,v,, v n ) in R n are called equal if u = v,u = v,, u n = v n The sum u + v is defined by u + v = (u +v, u +v,, u n +v n ) and if k is any scalar, the scalar multiple ku is defined by ku = (ku,ku,,ku n ) 05/6/4 Elementary Linear Algebra

12 4- Remarks The operations of addition and scalar multiplication in this definition are called the standard operations on R n. The zero vector in R n is denoted by 0 and is defined to be the vector 0 = (0, 0,, 0). If u = (u,u,,u n ) is any vector in R n, then the negative (or additive inverse) of u is denoted by -u and is defined by -u = (-u,-u,,-u n ). The difference of vectors in R n is defined by v u = v + (-u) = (v u,v u,,v n u n ) 05/6/4 Elementary Linear Algebra

13 Theorem 4.. (Properties of Vector in R n ) If u = (u,u,,u n ), v = (v,v,, v n ), and w = (w,w,, w n ) are vectors in R n 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 u = u 05/6/4 Elementary Linear Algebra 3

14 4- Euclidean Inner Product Definition If u = (u,u,,u n ), v = (v,v,, v n ) are vectors in R n, then the Euclidean inner product u v is defined by u v = u v + u v + + u n v n Example The Euclidean inner product of the vectors u = (-,3,5,7) and v = (5,-4,7,0) in R 4 is u v = (-)(5) + (3)(-4) + (5)(7) + (7)(0) = 8 05/6/4 Elementary Linear Algebra 4

15 Theorem 4.. Properties of Euclidean Inner Product If u, v and w are vectors in R n 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 05/6/4 Elementary Linear Algebra 5

16 4- Example (3u + v) (4u + v) = (3u) (4u + v) + (v) (4u + v ) = (3u) (4u) + (3u) v + (v) (4u) + (v) v =(u u) + (u v) + (v v) 05/6/4 Elementary Linear Algebra 6

17 4- Norm and Distance in Euclidean n-space We define the Euclidean norm (or Euclidean length) of a vector u = (u,u,,u n ) in R n by u / ( uu) u u... un Similarly, the Euclidean distance between the points u = (u,u,,u n ) and v = (v, v,,v n ) in R n is defined by d ( u, v) u v ( u v) ( u v)... ( u n vn) 05/6/4 Elementary Linear Algebra 7

18 4- Example 3 Example If u = (,3,-,7) and v = (0,7,,), then in the Euclidean space R 4 u () (3) ( ) (7) d( u, v) ( 0) (3 7) ( ) (7 ) 58 05/6/4 Elementary Linear Algebra 8

19 Theorem 4..3 (Cauchy-Schwarz Inequality in R n ) If u = (u,u,,u n ) and v = (v, v,,v n ) are vectors in R n, then u v u v 05/6/4 Elementary Linear Algebra 9

20 Theorem 4..4 (Properties of Length in R n ) If u and v are vectors in R n 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) 05/6/4 Elementary Linear Algebra 0

21 Theorem 4..5 (Properties of Distance in R n ) If u, v, and w are vectors in R n 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) 05/6/4 Elementary Linear Algebra

22 Hasil kali Titik dari Vektor Jika u dan v adalah vektor - vektor dalam ruang berdimensi 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 u 0 v cos q jika jika u 0 dan v 0 u 0 atau v 0

23 u.v = u.v + u.v +u 3.v 3 R 3 u.v = u.v + u.v R cosq u. v u. v CONTOH : u = (,-,) DAN v = (,,), CARILAH u.v dan tentukan sudut antara u dan v

24 Sudut Antar Vektor Jika u dan v adalah vektor-vektor tak nol, maka : cos q u.v u v

25 Hasil kali titik bisa digunakan untuk memperoleh informasi mengenai sudut antara 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 =/ jika dan hanya jika u.v=0

26 u.v = u.v + u.v +u 3.v 3 R 3 u.v = u.v + u.v R CONTOH : u = (,-,) dan v = (,,), Carilah u.v serta tentukan sudut antara u dan v

27 4- Orthogonality Two vectors u and v in R n are called orthogonal if u v = 0 Example 4 In the Euclidean space R 4 the vectors u = (-, 3,, 4) and v = (,, 0, -) are orthogonal, since u v = (-)() + (3)() + ()(0) + (4)(-) = 0 If u and v are called orthogonal, we writes: u v 05/6/4 Elementary Linear Algebra 7

28 Hasil Kali Silang Vektor Jika hasil kali titik berupa suatu skalar maka hasil kali silang berupa suatu vektor. Jika u = (u,u,u 3 ) dan v = (v,v,v 3 ) adalah vektor-vektor dalam ruang berdimensi 3, maka hasil kali silang u x v adalah vektor yang didefinisikan sebagai u x v =(u v 3 - u 3 v,u 3 v - u v 3,uv - u v ) atau dalam notasi determinan : u x v u v u v 3 3, u v u v 3 3, u v u v

29 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

30 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 = u v (u.v) 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

31 Vector - quick reminder Jika diketahui: v = (,0, -5) dan w = (3,,4), maka: v. w = v. w cos q (hasil kali titik - proyeksi) v x w (hasil kali silang) 05/6/4 Elementary Linear Algebra 3

32 Theorem 4..7 (Pythagorean Theorem in R n ) If u and v are orthogonal vectors in R n which the Euclidean inner product, then u + v = u + v 05/6/4 Elementary Linear Algebra 3

33 4- Matrix Formulae for the Dot Product If we use column matrix notation for the vectors u = [u u u n ] T and v = [v v v n ] T, or then u u u and v v n v n u v = v T u Au v = u A T v u Av = A T u v 05/6/4 Elementary Linear Algebra 33

34 4- Example 5 Verifying that Au v= u A t v 05/6/4 Elementary Linear Algebra , 4, v u A

35 05/6/4 Elementary Linear Algebra A Dot Product View of Matrix Multiplication If A = [a ij ] is an mr matrix and B =[b ij ] is an rn matrix, then the ijthe entry of AB is a i b j + a i b j + a i3 b 3j + + a ir b 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 r, r,, r m and the column vectors of B are c, c,, c n, n m m m n n AB c r c r c r c r c r c r c r c r c r

36 4- Example 6 A linear system written in dot product form system dot product form 05/6/4 Elementary Linear Algebra x x x x x x x x x 0 5 ),, ( (,5,-8) ),, ( (,-7,-4) ),, ( (3,-4,) x x x x x x x x x

37 Homework. Gunakan vektor-vektor untuk mencari cosinus sudut dibagian dalam sudut segitiga dengan titik-titik sudut (-, 0), (-, ) dan (, 4). Diketahui vektor u = (, -3, 4 ) dan v = ( -, 3, ). Berapakah nilai u x v? 3. Carilah luas segitiga yang ditentukan oleh titik-titik A (,, 0 ), B ( -, 0, ), C ( 0, 4, 3 ). 4. Misalkan u =(-, 3, ) w=(,, -). Cari semua vektor y yang memenuhi u x y = w! 05/6/4 Elementary Linear Algebra 37