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A Euclidean isometry f of a Euclidean space E defines a linear isometry of the associated vector space (by linear isometry, it is meant an isometry that is also a linear map) in the following way: denoting by Q – P the vector , (if O is an arbitrary point of E, one has
A linear form is a linear map from a vector space V over a field F to the field of scalars F, viewed as a vector space over itself. Equipped by pointwise addition and multiplication by a scalar, the linear forms form a vector space, called the dual space of V , and usually denoted V* [ 17 ] or V ′ .
In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted or . It is a geometric space in which two real numbers are required to determine the position of each point . It is an affine space , which includes in particular the concept of parallel lines .
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the ...
Vectors in a Euclidean space form a linear space, but each vector has also a length, in other words, norm, ‖ ‖. A real or complex linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space. Many spaces of sequences or functions are ...
A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied. [11] The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span
In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself.
In modern geometry, Euclidean spaces are often defined from linear algebra. More precisely, a Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals , and a group action of the additive group of , which is free and transitive (See Affine space for details of this construction).