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Applications of Matrix and Linear Transformation in Geometric and Computational Problems by Algebra Research Group Dept. of Mathematics Course 2.

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Presentasi berjudul: "Applications of Matrix and Linear Transformation in Geometric and Computational Problems by Algebra Research Group Dept. of Mathematics Course 2."— Transcript presentasi:

1 Applications of Matrix and Linear Transformation in Geometric and Computational Problems by Algebra Research Group Dept. of Mathematics Course 2

2 What do you see?

3

4 About Matrix Why Matrix ? Matrix Operations : Summation and Multiplications Invers Matrix Determinant

5 What you know All about (1x1) matrices OperationExampleResult o Addition o Subtraction5 – 14 o Multiplication2 x 24 o Division12 / 34

6 What you may guess Numbers can be organized in boxes, e.g.

7 Matrix Notation

8 Many Numbers

9 Matrix Notation

10 Useful Subnotation

11

12 Matrix Operations Addition Subtraction Multiplication Inverse

13 Addition

14 Addition

15 Addition Conformability To add two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B

16 Subtraction

17 Subtraction

18 Subtraction Conformability To subtract two matrices A and B: # of rows in A = # of rows in B # of columns in A = # of columns in B

19 Multiplication Conformability Regular Multiplication To multiply two matrices A and B: # of columns in A = # of rows in B Multiply: A (m x n) by B (n by p)

20 Multiplication General Formula

21 Multiplication I

22 Multiplication II

23 Multiplication III

24 Multiplication IV

25 Multiplication V

26 Multiplication VI

27 Multiplication VII

28 Inner Product of a Vector (Column) Vector c (n x 1)

29 Inverse A number can be divided by another number - How do you divide matrices? Note that a / b = a x 1 / b And that a x 1 / a = 1 1 / a is the inverse of a

30 Unany operations: Inverse Matrix ‘ equivalent ’ of 1 is the identity matrix Find A -1 such that A -1 * A = I

31 Unary Operations: Inverse

32 Inverse of 2 x 2 matrix Find the determinant = (a 11 x a 22 ) - (a 21 x a 12 ) For det( A ) = (2x3) – (1x5) = 1 o A determinant is a scalar number which is calculated from a matrix. This number can determine whether a set of linear equations are solvable, in other words whether the matrix can be inverted.

33 Inverse of 2 x 2 matrix Swap elements a 11 and a 22 Thus becomes

34 Inverse of 2 x 2 matrix Change sign of a 12 and a 21 Thus becomes

35 Inverse of 2 x 2 matrix Divide every element by the determinant Thus becomes (luckily the determinant was 1)

36 Inverse of 2 x 2 matrix Check results with A -1 A = I Thus equals

37 Outer Product of a Vector (Column) vector c (n x 1) Solving System of Linear Equtions

38 –3x + 2y – 6z = 6……(1) 5x + 7y – 5z = 6……(2) x + 4y – 2z = 8 …….(3) Using ELIMINATION of X

39 What is ellimination? Elementer Row Operations

40 Elementer Row Operations (ERO)

41 Elementer Row Operation (ERO)

42 Rewrite only the coefficient –3 x + 2 y – 6 z = 6 5 x + 7 y – 5 z = 6 x + 4 y – 2 z = 8

43 ERO only on the coefficient

44

45 Solving System of Linear Equations using Matrix –3 x + 2 y – 6 z = 6 5 x + 7 y – 5 z = 6 x + 4 y – 2 z = 8

46

47 Linear Transformation

48 Morphing is just a linear transformation between a base shape and a target shape.

49 Ruang Berdimensi 2 Ruang berdimensi 2 merupakan kumpulan titik-titik (vektor) berikut Anggota / elemen pada ruang berdimensi 2 disebut vektor dengan dua komponen.

50 Ruang Berdimensi 3 Ruang berdimensi 3 merupakan kumpulan titik-titik berikut Anggota / elemen pada ruang berdimensi 3 disebut vektor dengan tiga komponen.

51 Transformasi linear pada ruang dimensi 2 dan 3 Transformasi linear f adalah fungsi atau yang mempunyai sifat

52 Contoh transformasi linear pada ruang dimensi 2 Pencerminan terhadap sumbu x Proyeksi terhadap sumbu y Rotasi sebesar 90 derajat berlawanan arah dengan jarum jam

53 Matriks representasi pencerminan terhadap sumbu x (1) Diberikan fungsi berikut dengan definisi Namakan

54 Matriks representasi pencerminan terhadap sumbu x (2) Pemetaan tersebut dapat dinyatakan sebagai Dapat dicari bayangan titik P (2,4) ketika dicerminkan terhadap sumbu x sbb :

55 Matriks representasi proyeksi terhadap sumbu x (1) Didefinisikan proyeksi terhadap sumbu x di ruang berdimensi 3 sebagai berikut Namakan

56 Matriks representasi proyeksi terhadap sumbu x (2) Jadi proyeksi terhadap sumbu x di ruang berdimensi 3 dapat dinyatakan dengan Bayangan titik P (1,2,3) adalah

57 Gambar semula

58 Hasil transformasi (1)

59 Hasil transformasi (2)

60 Hasil transformasi (3)

61 Kesimpulan (1) Penggunaan Matriks dalam SPL Masalah/ProblemSPL Matriks Augmented Bentuk Eselon Baris tereduksi SPL Baru Solusi/ Penyelesaian

62 Kesimpulan (1) Penggunaan Matriks dalam SPL MasalahSistem Persamaan LinearMatriks yang diperluasBentuk eselon baris tereduksiPenyelesaian

63 Kesimpulan (2) Hubungan Transformasi Linear dan Matriks Setiap transformasi linear dapat diwakili oleh suatu matriks. Sebaliknya, suatu matriks dapat membangkitkan suatu transformasi linea r


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