Seminar First course 2022-2023
The row space of a matrix refers to the vector space spanned by its row vectors. It is a subspace of the ambient vector space and provides important insights into the properties and structure of the matrix. Here's a comprehensive reply to your question: The row space of a matrix A is the vector space spanned by its row vectors. In other words, it is the set of all possible linear combinations of the row vectors of A. The row space can be denoted as R (A) or Row (A). The row space is a subspace of the vector space in which the matrix is defined. For example, if A is an m-by-n matrix, the row space is a subspace of R^n, the n-dimensional Euclidean space. The dimension of the row space is equal to the rank of the matrix. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It can be computed using various methods, such as row reduction or singular value decomposition (SVD). The row space is closely related to other fundamental subspaces of a matrix, such as the column space, null space, and left null space. The row space and column space are orthogonal complements of each other. This means that the row space and column space have no non-zero vectors in common, and their intersection is the zero vector. The row space plays a crucial role in applications such as solving linear systems, finding bases for fundamental subspaces, determining the rank and nullity of a matrix, and understanding the geometry of linear transformations. To summarize, the row space of a matrix refers to the vector space spanned by its row vectors. It provides insights into the properties and structure of the matrix, and its dimension is equal to the rank of the matrix. The row space is closely related to other fundamental subspaces of the matrix and has various applications in linear algebra.
The column space of a matrix refers to the vector space spanned by its column vectors. In other words, it is the set of all possible linear combinations of the columns of the matrix. The column space is also known as the range or image of the matrix transformation. When we multiply a matrix A by a vector x, the result Ax can be expressed as a linear combination of the columns of A. Each entry of the vector x determines the scalar coefficient for the corresponding column vector of A. This geometric interpretation helps us understand that Ax lies in the subspace spanned by the columns of A. The column space of an m-by-n matrix A, denoted as Col(A), is a linear subspace of R^m. The dimension of the column space is called the rank of the matrix and is at most min(m, n). To summarize, the column space of a matrix represents the subspace spanned by its column vectors and provides valuable insights into the properties and transformations associated with the matrix.
The null space of a matrix refers to the set of all vectors that, when multiplied by the matrix, result in the zero vector. In other words, it is the solution space of the homogeneous equation Ax = 0, where A is the matrix and x is a vector.To find the null space of a matrix, you can follow these general steps:
Row-reduce the matrix to its reduced row-echelon form (RREF). This can be done using techniques such as Gaussian elimination. The row reduction does not change the augment of the matrix because the augment is always zero.
Write out the RREF matrix in equation form.
Reparameterize the free variables and solve for the unknowns. The free variables are variables that are not leading variables in the RREF matrix.
Rewrite the solution as a linear combination of vectors. The weights of the linear combination are the free variables. This allows you to express the null space as a span.
Seminar Second Course 2022-2023
In the seminar numerical solution for nonlinear equations have been studied. Ridders method have been discussed. An improvement of Ridders method with combination of Bisection and newton Raphson methods have been presented. An algorithm for the proposed method has been stated. Moreover, several examples are included to demonstrate the validity and applicability of the presented technique. MATLAB program involved for numerical computations. The proposed method applied for given examples. The error analysis table presents the obtained numerical results. The numerical solutions which found by MATLAB program has good results in terms of accuracy.
This paper presents an efficient and straightforward methodology with less computational complexities to title the bi-level objective linear fractional programming problem with fuzzy interval coefficients (BILOLFPP with FIC). To construct the methodology, the concept of mean technique is utilized to tackle the fuzzy numbers in addition to adding to α = [mean (ai) , mean (bi)] , i = 1, …, n, then. Accordingly, the fuzzy programming issue is converted into a single objective linear fractional programming problem (SOLFPP with FIC) by the utilize of weight function. The fuzzy technique has significant structural transform metamorphosis during the recent decades. Numerous to mention introduced have been undertaken to explanation fuzzy methodology for linear, non-linear programming issues. While, the previous finding that introduced have been conflicting, recent studies of competitive situations indicate that LFPP with fuzzy interval coefficients (LFPP with FIC) has an advantageous effect mostly on comparison situation. One of the suggestions which we found is interval approximations, closed interval approximation of sequential fuzzy number for resolving fuzzy number LFPP without changing it to a crisp issue. A new variant of modified simplex methodology is studied here just for resolving fuzzy number LFPP utilizing fuzzy arithmetic. Consequently, fuzzy representation of some important theories of fuzzy LFPP has been reproved. A fuzzy LFPP with FIC is worked out as numerical examples illustrate to the suggested methodology. On iterative processes, it decreases the overall processing time to explain, the modified simplex methodology for solving BILLFPP with FIC with out to crisp by taking numerical examples and compare with Nasseri, Verdegay and Mahmoudi methodology changing it to a crisp issue [9].
In this paper, a bi-level linear fractional programming problem (BILLFPP) with fuzzy interval coefficient (FIC) is contemplate where really all of it is coefficients in the goal function and constraints are fuzzy intervals (FIs). Firstly, to resolve this issue, we are going to construct two LFPP with fuzzy coefficients. Before all else, of these issues is a LFPP where all of coefficients are upper approximations of FIs and the other is a LFPP where all of coefficients are lower approximations of FIs. Secondly, the BILLFPP is transformed to the form of single goal LFPP and QFPP. We address problems with a factorised or non-factorised optimisation problem and homogeneous or non-homogeneous constraints. Our proposed technique is based on a mathematical model that converts the QFPP to a LPP by solving the problem in an algebraic expression with a Taylor series. This technique, which is based on the LPP solution, can be applied to specific problems. NLFPP containing nonlinear constraints, on iterative processes, it decreases the overall processing time. To explain, the novel technique for solving BILLFPP by taking numerical examples and compare with Jayalakshmi (2015) and Syaripuddin et al. (2017).
In this paper, a bi-level linear fractional programming problems (BILLFPP) with fuzzy interval coefficients (FIC) considered where all of it is coefficients in the goal function and constraints are fuzzy intervals (FIs). Firstly, to resolve this issue, we construct two LFPP with fuzzy coefficients one of these problems is a LFPP where all of it is coefficients are upper approximations of (FIs) and the other is a LFPP, where all of it is coefficients are lower approximations of (FIs) secondly, the BILLFPP are converted in to a single objective LFPP and QFPP. At the end, and we use game matrix technique which yields an efficient resolution of the original BILLFPP with fuzzy interval, the single objective LFPP or QFPP is resolved by modified simplex technique. Finally, numerical examples are given to demonstrate the results.