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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.
A.4.2 Using linear maps to solve linear systems. Encoding a linear system as a matrix equation is more than just a notational trick. Perhaps most fundamentally, the resulting linear map viewpoint can then be used to provide unparalleled insight into the exact structure of solutions to the original linear system.
Dec 26, 2022 · To get such information, we need to restrict to functions that respect the vector space structure — that is, the scalar multiplication and the vector addition. Functions with this property, which we’re going to define shortly, are called linear maps.
where each of the coefficients \(a_{ij} \) and \(b_i \) is in \(\mathbb{F} \). Linear maps and their properties give us insight into the characteristics of solutions to linear systems.
Feb 9, 2020 · In this lecture, we discuss the definition of a linear map, relating it to the definition of a vector space. We also give an elementary example, illustrating...
- 8 min
- 9.9K
- Kyle Broder
The set of linear maps L(V,W) is itself a vector space. For S,T ∈ L(V,W) addition is defined as (S +T)v = Sv +Tv for all v ∈ V. For a ∈ F and T ∈ L(V,W) scalar multiplication is defined as (aT)(v) = a(Tv) for all v ∈ V. You should verify that S + T and aT are indeed linear maps again and that all properties of a vector space are ...
Learn the definition, interpretation and representation of linear and affine maps, and how to approximate non-linear maps with them. See examples, applications and related concepts such as matrix inverses and norms.
