Math 2.0

definition

Linear isometry

Definition

Let VV and WW be normed vector spaces over R\R or C\C. A linear isometry from VV to WW is a linear map T ⁣:V→WT \colon V \to W with ∥T(v)∥=∥v∥for all v∈V.\norm{T(v)} = \norm{v} \qquad \text{for all } v \in V.

Remarks

A linear isometry is injective, because T(v)=0T(v) = 0 implies ∥v∥=0\norm{v} = 0. It keeps distances: ∥T(u)−T(v)∥=∥u−v∥\norm{T(u) - T(v)} = \norm{u - v}. Between inner product spaces, it keeps inner products, see the next statement. Between spaces of the same finite dimension, a linear isometry is bijective.

Explanations

L1 human

A linear isometry is a linear map that keeps all lengths. Rotations and reflections of the plane are examples. A map that keeps lengths cannot squeeze anything to zero.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 5 nodes in this order. The foundations come first, and the list ends with this node.

Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionNorm, definition, no evidence levelNormdefinitionNormed vector space, definition, no evidence levelNormed vector spacedefinitionLinear isometry, definition, no evidence levelLinear isometrydefinition
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Vector space definition
  2. Linear map definition
  3. Norm definition
  4. Normed vector space definition
  5. Linear isometry definition

What depends on this

1 node uses this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Isometry, Linear isometry

What links here

Used by