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Applications of Matrix and Linear
Transformationin
Geometric and Computational Problems
byAlgebra Research Group
Dept. of Mathematics
Course 2
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What do you see?
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About Matrix
• Why Matrix ?• Matrix Operations :
Summation and Multiplications• Invers • Matrix Determinant
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What you know• All about (1x1) matrices
• Operation Example Result
o Addition 2 + 2 4o Subtraction 5 – 1 4o Multiplication 2 x 2 4o Division 12 / 3 4
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What you may guess
• Numbers can be organized in boxes, e.g.
1
4
2
3
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Matrix Notation
A
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Many Numbers 31 23 16 99 08 12 14 73 85 98 33 94 12 75 02 57
92 75 11 28 39 57 17 38 18 38 65 10 73 16 73 77 63 18 56
18 57 02 74 82 20 10 75 84 19 47 14 11 84 08 47 57 58 49
48 28 42 88 84 47 48 43 05 61 75 98 47 32 98 15 49 01 38
65 81 68 43 17 65 21 79 43 17 59 41 37 59 43 17 97 65 41
35 54 44 75 49 03 86 93 41 76 73 19 57 75 49 27 59 34 27
59 34 82 43 19 74 32 17 43 92 65 94 13 75 93 41 65 99 13
47 56 34 75 83 47 48 73 98 47 39 28 17 49 03 63 91 40 35
42 12 54 31 87 49 75 48 91 37 59 13 48 75 94 13 75 45 43
54 32 53 75 48 90 37 59 37 59 43 75 90 33 57 75 89 43 67
74 73 10 34 92 76 90 34 17 34 82 75 98 34 27 69 31 75 93
45 48 37 13 59 84 76 59 13 47 69 43 17 91 34 75 93 41 75
90 74 17 34 15 74 91 35 79 57 42 39 57 49 02 35 74 23 57
75 11 35
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Matrix Notation
A
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Useful Subnotation
A2 2
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Useful Subnotation
A8 40
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Matrix Operations
• Addition• Subtraction• Multiplication• Inverse
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Addition
1
4
2
3
5
8
6
7+ =
A B+ =
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Addition
1
4
2
3
5
8
6
7+ =
6
12
8
10
A B+ = C
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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
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Subtraction
1
4
2
3
5
8
6
7- =
B A- =
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Subtraction
1
4
2
3
5
8
6
7- =
4
4
4
4
B A- = C
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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
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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)
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Multiplication General Formula
C ij = A ik x B kjk=1
n
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Multiplication I
1
4
2
3
5
8
6
7x =
A Bx =
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Multiplication II1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)
C 11 = A 11 x B 11k=1
n
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Multiplication III1
4
2
3
5
8
6
7x =
A Bx = C
(5x1)+ (6x3)
C 11 = A 12 x B 21k=2
n
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Multiplication IV1
4
2
3
5
8
6
7x =
A Bx = C
23 (5x2)+ (6x4)
C 12 = A 1k x B k2k=1
n
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Multiplication V1
4
2
3
5
8
6
7x =
A Bx = C
23
(7x1)+ (8x3)
34
C 21 = A 2k x B k1k=1
n
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Multiplication VI1
4
2
3
5
8
6
7x =
A Bx = C
23 34
(7x2)+ (8x4)31
C 22 = A 2k x B k2k=1
n
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Multiplication VII
1
4
2
3
5
8
6
7x =
A Bx = C
23 34
31 46
m x n n x p m x p
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Inner Product of a Vector
• (Column) Vector c (n x 1)
c' c
2 14 x 2
1
4=
=
=
c' c21
=c 2
1
4
(2x2)+(4x4)+(1x1)
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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
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Unany operations: Inverse
• Matrix ‘equivalent’ of 1 is the identity matrix
• Find A-1 such that A-1 * A = I
1
1
0
0=I
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Unary Operations: Inverse
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Inverse of 2 x 2 matrix• Find the determinant = (a11 x a22) - (a21 x a12)
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.
2
3
5
1=A
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Inverse of 2 x 2 matrix• Swap elements a11 and a22
Thus
becomes
2
3
5
1=A
3
2
5
1
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Inverse of 2 x 2 matrix• Change sign of a12 and a21
Thus
becomes
3
2
5
1=A
3
2
-5
-1
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Inverse of 2 x 2 matrix• Divide every element by the determinant
Thus
becomes
(luckily the determinant was 1)
3
2
-5
-1=A
3
2
-5
-1
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Inverse of 2 x 2 matrix• Check results with A-1 A = I
Thus
equals
3
2
-5
-1x
1
1
0
0
2
3
5
1
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Outer Product of a Vector
• (Column) vector c (n x 1)
c c'
2 14x2
1
4
=
c c'
=c 2
1
4
4 8 2
8 16 4
2 4 1
Solving System of Linear Equtions
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Solving System of Linear Equtions
–3x + 2y – 6z = 6……(1)
5x + 7y – 5z = 6……(2) x + 4y – 2z = 8 …….(3)
Using ELIMINATION of X
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What is ellimination?
Elementer Row Operations
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Elementer Row Operations
(ERO)
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Elementer Row Operation
(ERO)
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Rewrite only the coefficient
–3x + 2y – 6z = 6 5x + 7y – 5z = 6 x + 4y – 2z = 8
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ERO only on the coefficient
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Solving System of Linear Equations
using Matrix –3x + 2y – 6z = 6
5x + 7y – 5z = 6 x + 4y – 2z = 8
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Linear Transformation
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Linear TransformationMorphing is just a linear transformation between a base shape and a target shape.
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Ruang Berdimensi 2 • Ruang berdimensi 2 merupakan kumpulan titik-
titik (vektor) berikut
• Anggota / elemen pada ruang berdimensi 2 disebut vektor dengan dua komponen.
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Ruang Berdimensi 3• Ruang berdimensi 3 merupakan kumpulan titik-
titik berikut
• Anggota / elemen pada ruang berdimensi 3 disebut vektor dengan tiga komponen.
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Transformasi linear pada ruang dimensi 2 dan 3
Transformasi linear f adalah fungsi atau yang mempunyai sifat
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Contoh transformasi linear pada ruang dimensi 2
• Pencerminan terhadap sumbu x• Proyeksi terhadap sumbu y• Rotasi sebesar 90 derajat
berlawanan arah dengan jarum jam
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Matriks representasi pencerminan terhadap sumbu x
(1)• Diberikan fungsi berikut
dengan definisi
Namakan
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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 :
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Matriks representasi proyeksi terhadap sumbu x (1)• Didefinisikan proyeksi terhadap sumbu x di ruang
berdimensi 3 sebagai berikut
• Namakan
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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
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Gambar semula
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Hasil transformasi (1)
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Hasil transformasi (2)
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Hasil transformasi (3)
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Kesimpulan (1)Penggunaan Matriks dalam SPL
Masalah/Problem SPLMatriks
Augmented
BentukEselonBaris
tereduksi
SPL BaruSolusi/
Penyelesaian
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Kesimpulan (1)Penggunaan Matriks dalam SPL
Masalah
Sistem Persamaan Linear
Matriks yang diperluas
Bentuk eselon baris tereduksi
Penyelesaian
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Kesimpulan (2)Hubungan Transformasi
Linear dan Matriks
• Setiap transformasi linear dapat diwakili oleh suatu matriks.
• Sebaliknya, suatu matriks dapat membangkitkan suatu
transformasi linear