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Then L(V,W) is a subspace of W V since it is closed under addition and scalar multiplication. Note that L(F n,F m) can be identified with the space of matrices F m×n in a natural way. In fact, by choosing appropriate bases for finite-dimensional spaces V and W, L(V,W) can also be identified with F m×n. This identification normally depends on ...
Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
The simplest example of a vector space over a field F is the field F itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all n -tuples (sequences of length n ) ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\dots ,a_{n})} of elements a i of F form a vector space that ...
Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication; Flat (geometry), a Euclidean subspace; Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector space
In mathematics, projectivization is a procedure which associates with a non-zero vector space V a projective space P(V), whose elements are one-dimensional subspaces of V.More generally, any subset S of V closed under scalar multiplication defines a subset of P(V) formed by the lines contained in S and is called the projectivization of S.
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that () is the intersection of the closed sets containing X. This equivalence remains true for partially ordered sets with the greatest-lower-bound property , if one replace "closed sets" by "closed elements" and ...
An example of an external operation is scalar multiplication, where a vector is multiplied by a scalar and result in a vector. An n -ary multifunction or multioperation ω is a mapping from a Cartesian power of a set into the set of subsets of that set, formally ω : X n → P ( X ) {\displaystyle \omega :X^{n}\rightarrow {\mathcal {P}}(X)} .
Homogeneity of degree 1 / operation of scalar multiplication () = Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.