Aljabar Linier Pertemuan 1.pptxjjtnrntriotno

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1. Linear Algebra By : Elvira Wahyu Arum Fanani, S.Pd ., M.T. Linier Algebra 1

2. Robot lengan mewakili bidang mekanika & robotika di teknik mesin, di mana linear algebra digunakan untuk: menghitung pergerakan (kinematika), mengatur gaya dan torsi, mengontrol posisi ujung lengan saat mengambil/memindahkan beban. Introduction to Linear Algebra in Engineering

3. Pendahuluan dan Sistem Persamaan Linear (1-2) Determinan (3-4) Vektor (5-6) Ruang Vektor Euclid (7) Ruang Vektor Umum (9-10) Ruang Hasil Kali Dalam (11-12) Nilai Eiger dan Vektor Eigen (13) Transformasi Linier (14-15) Topics

4. Kontrak Kuliah

5. Sub-Topics Introduction to Linear Equation Systems Gauss–Jordan Elimination Matrices and Matrix Operations Matrix Algebra, Inverse Matrix Elementary Matrices: How to Find the Inverse Matrix Types of Matrices

6. Systems of Linear Equations There is a line in the plane There is a line in the plane These two lines are examples of Linear Equations. In general, a linear equation has infinite variables:   General form of linear equations   Where: and are coefficients is a constant or are variables/unknown  

7. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS Exercise 1 Are the following equations linear to ?  

8. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS A system of linear equations is a system consisting of more than one linear equation. The following is the general form of a linear equation: + …….+ + …….+ + …….+ + …….+   Examples:  

9. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS The values of the variables that satisfy a system of linear equations are called solutions to a linear equation. In general, the number of solutions to a System of Linear Equations (SLE) / Sistem Persamaan Linier (SPL) in Indonesian is as follows: 0 Solution / no solution 1 Solution Infinitely many Solutions An SLE that has no solutions is said to be inconsistent , while an SLE that has solutions is said to be consistent.

10. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS Geometrically, the SLE solution can be described as follows:

11. INTRODUCTION TO SYSTEMS OF LINEAR EQUATIONS Exercise 2 Determine whether the following SLE has 1 solution, multiple solutions, or no solutions! 1.

12. A German mathematician, he discovered Gaussian elimination , which is used to obtain Row Echelon Matrices / Matriks Eselon Baris A German geodesist, he developed Gaussian elimination into Gauss-Jordan elimination to obtain Reduced Row Echelon Matrices / Matriks Eselon Baris Tereduksi Gauss–Jordan Elimination 13

13. Baris nol : baris yang seluruh angkanya Adalah 0 Baris- tak - nol : baris yang mempunyai minimal nilai 1 angka tak nol Matriks esellon baris : Jika suatu baris adalah baris tak nol , maka angka tak nol pertama di baris tersebut harus angka 1. angka 1 ini disebut satu-utama (leading 1) . Baris nol harus dikelompokkan di dasar matriks . 3. Dalam 2 baris tak nol yang berurutan , satu utama dalam baris yang lebih bawah harus terletak di sebelah kanan dari satu utama baris yang lebih atas Matriks eselon baris tereduksi : memenuhi sifat 1,2, dan 3 ditambah 4. masing-masing kolom yang berisi satu utama mempunyai angka 0 di baris lainnya dalam kolom tersebut. Gauss–Jordan Elimination

14. Gauss–Jordan Elimination

15. Gauss Jordan Elimination Some a u th o rs use the t e rm Ga u ssian eli m ination to ref e r o n ly to the pr o ce d ure u n til the matrix is in e ch e lon form, a n d use t h e term Gau ss -Jordan el i mination to refer to the proc e d u re which en d s in reduc e d e c helon for m . In line a r alg e bra, Ga u s s – Jord a n e lim i n a tion is an alg o rithm for g e tting matrices in re d uc e d r ow echel o n form using ele m ent a ry r ow o p era t io n s. It is a v a riation of Ga u ssian eliminatio n .

16. Example 1 x + y + 2x + 3y 4x + 5 z z = 5 + 5z = 8 = 2

17. 5 2  Solution : fo l l o win g . The augmented m atrix of the system is the 1 2 4 1 3 1 |5 5 |8 5 |2  W e w i ll now p erf o rm row op e rat i ons u n til we o bt a in a matr i x in r e du c ed row ec h el o n form. 1 2 4 1 3 1 |5 5 |8 5 |2 1 1 1 4 5 1 3 -2

18. R 3-4 R 1 1 1 1 5 1 3 -2 4 5 2 1 1 1 5 1 3 -2 -4 1 -18

19. R 3 +4R2 1 1 1 5 1 3 -2 -4 1 -18 1 1 1 5 1 3 -2 13 -26

20. (1/ 1 3) R 3 1 1 1 5 1 3 -2 13 -26 1 1 1 5 1 3 -2 1 -2

21. R2-3R3 1 1 1 5 1 3 -2 1 -2 1 1 1 5 1 -2 1 4

22. 1 1 1 5 1 4 1 -2 1 4 1 -2 1 1 7 R 1- R 3

23. 1 1 7 1 4 1 -2 1 4 1 -2 1 3 R1 -R2

24.  F r om t hi s fi n a l mat ri x, s yst e m. It is we c an r e a d t h e s o l u ti on of t he X= 3 Y=4 Z=-2  Substitute x , y, z in system equation, if the right side = left side in three equation then the sol. is correct 3+4-2=5 2*3+3*4+5*(-2)=8 4*3+5*(-2)=2

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