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Linear algebra, a branch of mathematics dealing with vector spaces and linear mappings between these spaces, plays a critical role in various engineering disciplines, including fluid mechanics, fluid dynamics, and thermal energy systems. Its application in these fields is multifaceted and indispensable for solving complex problems.
In mathematics, the term linear is used in two distinct senses for two different properties: . linearity of a function (or mapping);; linearity of a polynomial.; An example of a linear function is the function defined by () = (,) that maps the real line to a line in the Euclidean plane R 2 that passes through the origin.
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line , that is, a polynomial function of degree zero or one. [ 2 ]
In mathematics, a linear equation is an equation that may be put in the form + … + + =, where , …, are the variables (or unknowns), and ,, …, are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation and may be arbitrary expressions , provided they do not contain any of the variables.
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector axioms .
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or | A |. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix.
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .