ALJABAR LINEAR ELEMENTER – Ebook written by VERSI APLIKASI. Read this book using Google Play Books app on your PC, android, iOS devices. Sistem Informasi. Aljabar Linear Elementer Versi Aplikasi Jilid 2 Edisi 8. Share to: Facebook; Twitter; Google; Digg; Reddit; LinkedIn; StumbleUpon. Anton. Buy Aljabar Linear Elementer Versi Aplikasi Ed 8 Jl 1 in Bandung,Indonesia. Get great deals on Books & Stationery Chat to Buy.

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Tidak meninggalkan sampah di ruangan kelas 6. Essential Linear Algebra with Applications. Elementary Linear Algebra, 9th Edition. Department Mathematics, Linkoping University. A matrix is a aljsbar array of numbers.

The numbers in the aplikadi are called the entries in the matrix. If m and n are positive integers then by a matrix of size m by n, or an m x n matrix, we shall mean a rectangular array consisting of mn numbers in a boxed display consisting of m rows and n columns.

Two matrices are defined to be equal if they have the same size and their corresponding entries are equal. Matrices of different sizes cannot be added or subtracted.

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If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of the matrix A by c. The matrix cA is said to be a scalar multiple of A. If A is an m x r matrix and B is an r x n matrix, then the product AB is the m x n matrix whose entries are determined as follows.

To find the entry in row i and column j of ABsingle out row i from the matrix A and column j from the matrix B. Multiply the corresponding entries from the row and column together, and then add up the resulting products.

If A is any m x n matrix, then the transpose of A, denoted by AT is defined to be the n x m matrix that results from interchanging the rows and columns of A ; that is, the first column of AT is the first row of A, the second column of AT is the second row of A, and so forth. If A is a square matrix, then the trace of A, denoted by tr Alpikasiis defined to be the sum of the entries on the main diagonal of A.


The trace of A is undefined if A is not a square matrix. The various costs in whole dollars involved in producing a single item of a product are given in the table: P Q R Material 1 2 1 Labor 3 2 2 Overheads 2 1 2 The numbers of items produced in one month at the four locations are linead follows: Let C be the “cost” matrix formed by the first apllikasi of data and let N be the matrix formed by the second set of data.

If no such matrix B can be found, then A is said to be singular. To find the inverse of an invertible matrix A, we must find a sequence of elementary row operations that reduces A to the identity and then perform this same sequence of operations on In to obtain A Adjoin the identity matrix to the right side of A, thereby producing a matrix of the form [A I], apply row operations to this matrix until the left side is reduced to I; these operations will convert the right side to A-1, so the final matrix will have the form [I A-1].

More generally, we define the determinant of an n x n matrix to be This method of evaluating det A is called cofactor expansion along the first row of A. Adjoint of Matrix If A is any n x n matrix and Cij is the cofactor of aijthen the matrix Is called the matrix of cofactor from A.

Solution Howard Anton System of Linear Equations Libear Anton Solution Consider a general system of two linear equations in the unknowns x and y: Augmented Matrices A system of m linear equations in n unknowns can be abbreviated by writing only aljwbar rectangular array of numbers This is called the augmented matrix for the system. This new system is generally obtained in a series of steps by applying the following three types of operations to eliminate unknowns systematically: Multiply an equation through by a nonzero constant.

Add a multiple of one equation to another. Method for Solving a System of Linear Equations Since the rows horizontal lines of an augmented matrix correspond to the equations in the associated versu, these three operations correspond to the following operations on the rows of the augmented matrix: Multiply a row through by a nonzero constant. Add a multiple of one row to another row. Gaussian Elimination Howard Anton To be of reduced row-echelon form, a matrix must have the following properties: If a row does not consist veri of zeros, then the first nonzero number in the row is a 1.

We call this a leading 1. If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.

In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row.


Each column that contains a leading 1 has zeros everywhere else in that column. A matrix that has the first three properties is said to be in row-echelon form. Thus, a matrix in reduced row-echelon form is of necessity in row-echelon form, but not conversely.

A system of linear equations is said to be homogeneous if the constant terms are all zero, the system has the form: This solution is called the trivial solution.

A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.

Special case In the special case of a homogeneous linear system of two equations in two unknowns, say: The graphs of the equations are lines through the origin, and the trivial solution corresponds to the points of intersection at the origin.

ALJABAR LINEAR | Reny Rian Marliana –

Position the vector w so that its initial point coincides with the terminal point of v. Vectors in Coordinate Systems If equivalent vectors, v and w, are located so that their initial points fall at the origin, then it is obvious that their terminal points must coincide since the vectors have the same length and direction ; thus the vectors have the same components. Conversely, vectors with the same components are equivalent since they have the same length and the same direction.

If, as shown in Figure 3. The set of all ordered n- tuples is called n-space and is denoted by Rn.


Note A set S with two or more vectors is: It is denoted by: In addition, we shall regard the zero vector space to be finite dimensional. The dimension of a finite-dimensional vector space V, denoted by dim Vis defined to be lineaf number of vectors in a basis for V.

In addition, we define the zero vector space to have dimension zero. We shall call linear transformations from Rn to Rm matrix transformations, since they can be carried out by matrix multiplication. To see that T is linear, observe that: Remember me on this computer. Enter the email address you signed up with and we’ll email you a reset link. Click here to sign up. Help Center Find new research papers in: