Linear Algebra Notes

By Xah Lee. Date: .

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in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W. That is, in set-builder notation,

In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by “collapsing” N to zero vector. The space obtained is called a quotient space and is denoted V/N (read V mod N or V by N).


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:

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

and if:

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

vector bundle dot product inner product inner product space

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