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Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
Let C be a set in a real or complex vector space. C is star convex (star-shaped) if there exists an x 0 in C such that the line segment from x 0 to any point y in C is contained in C. Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
The sizeof operator on such a struct gives the size of the structure as if the flexible array member were empty. This may include padding added to accommodate the flexible member; the compiler is also free to re-use such padding as part of the array itself. [2]
A linear subspace or vector subspace W of a vector space V is a non-empty subset of V that is closed under vector addition and scalar multiplication; that is, the sum of two elements of W and the product of an element of W by a scalar belong to W. [10] This implies that every linear combination of elements of W belongs to W. A linear subspace ...
The empty set is the unique initial object in Set, the category of sets.Every one-element set is a terminal object in this category; there are no zero objects.. Similarly, the empty space is the unique initial object in Top, the category of topological spaces and every one-point space is a terminal object in thi
In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, [4] a seminorm in a ...
A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient (or strictly convex) otherwise. [25] [26] A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone C is salient if and only if C ∩ −C ⊆ {0}.
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.