Math 2.0

Linear and multilinear algebra; matrix theory Matrix equations and groups Object

Special linear group

Builds on DeterminantGeneral linear group

Description

Let FF be a field and let n≥1n \ge 1. The special linear group is the set SLn(F)={A∈GLn(F):det⁡A=1}SL_n(F) = \{A \in GL_n(F) : \det A = 1\} of n×nn \times n matrices over FF with determinant 11, with matrix multiplication. It is a subgroup of the general linear group GLn(F)GL_n(F).

Remarks

It is the kernel of the group homomorphism det⁡ ⁣:GLn(F)→F×\det \colon GL_n(F) \to F^\times, so it is a normal subgroup. The quotient GLn(F)/SLn(F)GL_n(F) / SL_n(F) is isomorphic to F×F^\times. Over ℝ\R, its elements are the linear maps that keep volume and orientation. SL2(ℤ)SL_2(\Z) is the analogous group of integer matrices with determinant 11.

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General linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobject
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  1. General linear group object
  2. Identity matrix object
  3. Matrix definition
  4. Vector space definition
  5. Square matrix definition
  6. Linear map definition
  7. Multilinear map definition
  8. Alternating multilinear map definition
  9. Determinant definition
  10. Special linear group object

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Further reading

  1. Web https://en.wikipedia.org/wiki/Special_linear_group, lead section
About this page
Tag0V9PMN0RBH
URIm2://0V9PMN0RBH
Kindobject
WritingW1 stated
MSC15A15, 20G15
Instance ofGroup
UsesDeterminant, General linear group
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What this page needs for W2

  • section Invariants: a table of the main invariants (0 of 1)

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