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For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or generated by S; S is a generator set or a generating set of V.
This article uses the convention that vectors are denoted in a bold font (e.g. a 1), and scalars are written in normal font (e.g. a 1). The dot product of vectors a and b is written as a ⋅ b {\displaystyle \mathbf {a} \cdot \mathbf {b} } , the norm of a is written ‖ a ‖, the angle between a and b is denoted θ .
For any vector space V, the projection X × V → X makes the product X × V into a "trivial" vector bundle. Vector bundles over X are required to be locally a product of X and some (fixed) vector space V: for every x in X, there is a neighborhood U of x such that the restriction of π to π −1 (U) is isomorphic [nb 11] to the trivial bundle ...
rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:
To show that ad − bc is the signed area, one may consider a matrix containing two vectors u ≡ (a, c) and v ≡ (b, d) representing the parallelogram's sides. The signed area can be expressed as |u| |v| sin θ for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component ...
The dual of an infinite-dimensional space has greater dimension (this being a greater infinite cardinality) than the original space has, and thus these cannot have a basis with the same indexing set. However, a dual set of vectors exists, which defines a subspace of the dual isomorphic to the original space.
Thus Ax = 0 if and only if x is orthogonal (perpendicular) to each of the row vectors of A. It follows that the null space of A is the orthogonal complement to the row space. For example, if the row space is a plane through the origin in three dimensions, then the null space will be the perpendicular line through the origin.
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.