Math 2.0

definition

Orientation of a real vector space

Definition

Let VV be a real vector space of finite dimension n≥1n \ge 1. Two ordered bases BB and B′B' of VV have the same orientation when the change of basis matrix from BB to B′B' has positive determinant. This is an equivalence relation with exactly two classes. An orientation of VV is a choice of one of these two classes.

Remarks

A vector space with a chosen orientation is oriented, and the bases in the chosen class are positively oriented. The standard orientation of Rn\R^n is the class of the standard basis. In R2\R^2, exchanging the two basis vectors reverses the orientation. An invertible linear map keeps the orientation if and only if its determinant is positive.

Explanations

L1 human

An orientation fixes what counts as turning to the left, or as a right-handed system. In the plane, a counterclockwise pair of basis vectors and a clockwise pair have opposite orientations. A mirror reflection changes the orientation, and a rotation keeps it.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionSpanning set, definition, no evidence levelSpanning setdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionBasis, definition, no evidence levelBasisdefinitionDeterminant, definition, no evidence levelDeterminantdefinitionCoordinate vector, definition, no evidence levelCoordinate vectordefinitionChange of basis matrix, definition, no evidence levelChange of basis matrixdefinitionOrientation of a real vector space, definition, no evidence levelOrientation of a real…definition
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.
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Linear combination definition
  6. Linear map definition
  7. Linear subspace definition
  8. Linear independence definition
  9. Linear span definition
  10. Multilinear map definition
  11. Spanning set definition
  12. Alternating multilinear map definition
  13. Basis definition
  14. Determinant definition
  15. Coordinate vector definition
  16. Change of basis matrix definition
  17. Orientation of a real vector space definition

Sources

  1. Web https://en.wikipedia.org/wiki/Orientation_(vector_space), Definition

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