Linear Algebra Notes

By Xah Lee. Date: 2017-09-16.

Linear map

a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

An important special case is when V = W, in which case the map is called a linear operator,[1] or an endomorphism of V. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not.

A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension);[2] for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring.

Definition:

let u and v be members in vector space V.

let α be a member in field.

let f and g be function. Domain is V, codomain is vector space W.

let ⊕ be vector additon.

let * be multiplication for the field

let + be addition for the field

f(u ⊕ v) = f(u) ⊕ f(v)

f(α * u) = α * f(u)

Linear functional

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

dual space

any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.

definition

let V be a vector space, over the field {K, +, *}.
let a and b members of K.
let λf be a linear fuctional in V.
let λg be a linear fuctional in V.
let λ+ be addition defined for linear fuctionals.
let λ* be a scalar multiplication defined for linear fuctionals. i.e. 「a λ* λf = b」 for some a and b in the field.

and if:

(λf λ+ λg)[a] == λf [ a ] + λf [ b ]
(a λ* λf)[ a ] == a * λf [ a ]

Then, the set of all linear functionals of V, written as V* , with operators λ+ and λ* , are the dual space of V.

functional

In linear algebra, it refers to a linear mapping from a vector space V into its field of scalars

dot product

[ Dot product ] [ 2018-11-21 https://en.wikipedia.org/wiki/Dot_product ]

Inner product space

[ Inner product space ] [ 2018-11-21 https://en.wikipedia.org/wiki/Inner_product_space ]

vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (example: a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

[ Vector bundle ] [ 2018-11-21 https://en.wikipedia.org/wiki/Vector_bundle ]