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

What do you see?

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

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

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

Matrix Notation

Many Numbers

Matrix Notation

Useful Subnotation

Matrix Operations Addition Subtraction Multiplication Inverse

Addition

Addition

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

Subtraction

Subtraction

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

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)

Multiplication General Formula

Multiplication I

Multiplication II

Multiplication III

Multiplication IV

Multiplication V

Multiplication VI

Multiplication VII

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

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

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

Unary Operations: Inverse

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.

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

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

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

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

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

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

What is ellimination? Elementer Row Operations

Elementer Row Operations (ERO)

Elementer Row Operation (ERO)

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

ERO only on the coefficient

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

Linear Transformation

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

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

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

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

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

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

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 :

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

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

Gambar semula

Hasil transformasi (1)

Hasil transformasi (2)

Hasil transformasi (3)

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

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

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