If X1, X2 , ... , Xn are matrices of the same dimensions and a1, a2, ... an are constants, then
Y = a1*X1 + a2*X2 + ... + an,*Xn is a linear combination of the X matrices.
Linear Algebra
No. For example, linear algebra, for example, is about linear equations where the domain and range are matrices, not simple numbers. These matrices may themselves contain numbers that are real or complex so that not only is the range not the real numbers, but it is not based on real numbers either.
If you have a system, which can be expressed as a set of linear equations, then you can utilize matrices to help solve it. One example is an electrical circuit which uses linear devices (example are constant voltage sources and resistive loads). To find the current through each device, a set of linear equations is derived.
Yes, interchanging rows is permitted when solving a system of linear equations using matrices. This operation, known as row swapping, is one of the elementary row operations that can be performed during row reduction or when using methods like Gaussian elimination. It helps in simplifying the matrix and does not affect the solution of the system. Thus, it is a valid step in manipulating matrices.
Matrices are used in pretty much any situation where several linear equations are involved. They're used all over the place in physics, chemistry, engineering, and economics. I linked a site below with more information.
Toshinori Munakata has written: 'Matrices and linear programming with applications' -- subject(s): Linear programming, Matrices 'Solutions manual for Matrices and linear programming'
Inverse matrices are defined only for square matrices.
Linear Algebra
Matrices are tools to solve linear equations. Engineers use matrices in solving electrical problems in circuits using Thevenin's and Norton's theories.
Arthur Cayley
No. For example, linear algebra, for example, is about linear equations where the domain and range are matrices, not simple numbers. These matrices may themselves contain numbers that are real or complex so that not only is the range not the real numbers, but it is not based on real numbers either.
If you have a system, which can be expressed as a set of linear equations, then you can utilize matrices to help solve it. One example is an electrical circuit which uses linear devices (example are constant voltage sources and resistive loads). To find the current through each device, a set of linear equations is derived.
H. Neill has written: 'Vectors matrices and linear equations'
Charles Gordon Cullen has written: 'Matrices and linear transformations'
Zygmunt Dowgird has written: 'Krakowiany i ich zastosowanie w mechanice budowli' -- subject(s): Algebras, Linear, Linear Algebras, Matrices
Kent Franklin Carlson has written: 'Applications of matrix theory to systems of linear differential equations' -- subject(s): Differential equations, Linear, Linear Differential equations, Matrices
Linear Algebra is a branch of mathematics that enables you to solve many linear equations at the same time. For example, if you had 15 lines (linear equations) and wanted to know if there was a point where they all intersected, you would use Linear Algebra to solve that question. Linear Algebra uses matrices to solve these large systems of equations.