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A linear isometry also necessarily preserves angles, therefore a linear isometry transformation is a conformal linear transformation. Examples A linear map from C n {\displaystyle \mathbb {C} ^{n}} to itself is an isometry (for the dot product ) if and only if its matrix is unitary .
It is a linear isometry at the tangent space of every point on (), that is, it is an isometry on the infinitesimal patches. It preserves the curvature tensor at the tangent space of every point on B r ( x ) {\displaystyle B_{r}(x)} , that is, it preserves how the infinitesimal patches fit together.
If and are two finite-dimensional normed spaces with the same dimension, let (,) denote the collection of all linear isomorphisms :. Denote by ‖ ‖ the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors.
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if E is a Banach space with trivial centralizer and X and Y are compact, then every linear isometry of C(X; E) onto C(Y; E) is a strong Banach–Stone map.
A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. [18]
Any element of E(n) is a translation followed by an orthogonal transformation (the linear part of the isometry), in a unique way: (+) where A is an orthogonal matrix or the same orthogonal transformation followed by a translation: x ↦ A x + c , {\displaystyle x\mapsto Ax+c,} with c = Ab
Bahasa Indonesia ; Íslenska; Italiano ... It is a quasi-isometry if in addition it is ... An unusual property of normed vector spaces is that linear transformations ...
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.