concepts in linear algebra
The aim of this entry is to present a list of the key objects and operators used in linear algebra^{}. Each entry in the list links (or will link in the future) to the corresponding PlanetMath entry where the object is presented in greater detail. For convenience, this list also presents the encouraged notation to use (at PlanetMath) for these objects.
Some of this notation is simply an example of more general notation, either notation in set theory^{} or notation for functions. Some notation is also standard from category theory^{}.
Suppose $V$ is a vector space^{} over a field $K$. Where the field $K$ is clear from context it is sometimes eliminated from the notation. Let $L$ be a linear operator, or linear transformation, from $V$ to $W$, and $E$ be an endomorphism^{} of $V$.

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$\mathcal{L}(V,W)$, the set of linear transformations between vector spaces $V$ and $W$ (also ${\mathrm{Hom}}_{K}(V,W)$),

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basis of a vector space and the matrix associated with the basis,

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${dim}_{K}V$, dimension^{} (http://planetmath.org/Dimension2) of $V$,

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$\mathrm{span}\{{e}_{1},\mathrm{\dots},{e}_{n}\}$ vector space spanned by vectors $\{{e}_{i}\}$ (note that the list of vectors need not be finite). Some other notations are $({e}_{1},{e}_{2},\mathrm{\dots},{e}_{n})$ or $\u27e8{e}_{1},{e}_{2},\mathrm{\dots},{e}_{n}\u27e9$ (do not confuse last one with similar^{} inner product notation),

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$detE$, determinant^{} of a linear operator,

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$\mathrm{tr}E$, trace of a linear operator (also $\mathrm{trace}E$),

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$\mathrm{im}L$, image of a linear operator (also $\mathrm{img}L$ and $L(V)$),

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$\mathrm{ker}L$, kernel of a linear operator,

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for sets $A,B$ and a point $x\in V$, the expressions $A+B$, $AB$, $A+x$ are the Minkowski sums^{} (http://planetmath.org/MinkowskiSum2) (not especially if $A$ and $B$ are subspaces^{}, then their sum is a subspace, but the result may not be a direct sum^{}),

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${V}^{\ast}$, dual space^{} of a vector space $V$ (also ${V}^{\vee}$ or ${\mathrm{Hom}}_{K}(V,K)$),

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${A}^{\ast}$, adjoint operator of a linear operator,

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$V\oplus W$, direct sum of vector spaces $V$ and $W$ (both internal end external),

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$V{\otimes}_{K}W$, tensor product^{} of $V$ and $W$, and

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$V\wedge W$, antisymmetrized tensor product (also called the wedge product^{}),

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bilinear forms^{} and quadratic forms^{}

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generalizations^{} of vector spaces over a field to vector spaces over a division ring to modules over a ring.
Title  concepts in linear algebra 

Canonical name  ConceptsInLinearAlgebra 
Date of creation  20130322 14:13:30 
Last modified on  20130322 14:13:30 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  13 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 1600 
Classification  msc 1300 
Classification  msc 2000 
Classification  msc 1500 