Math 2.0

example

The special linear group is the kernel of the determinant

Example

Let n≥1n \ge 1. The determinant is a homomorphism det⁡:GLn(R)→R×\det : \mathrm{GL}_n(\mathbb{R}) \to \mathbb{R}^\times, because det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B). Here GLn(R)\mathrm{GL}_n(\mathbb{R}) is the general linear group, and R×\mathbb{R}^\times is the group of nonzero real numbers under multiplication. The kernel of det⁡\det is the set of matrices with determinant 11. This is the special linear group SLn(R)\mathrm{SL}_n(\mathbb{R}). Thus SLn(R)\mathrm{SL}_n(\mathbb{R}) is a normal subgroup of GLn(R)\mathrm{GL}_n(\mathbb{R}).

Remarks

For n≥2n \ge 2, the group GLn(R)\mathrm{GL}_n(\mathbb{R}) is not abelian. Thus the normality does not come from commutativity.

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To read this node, read these 14 nodes in this order. The foundations come first, and the list ends with this node.

General linear group, object, no evidence levelGeneral linear groupobjectGroup, concept, no evidence levelGroupconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionMultilinear 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 groupobjectThe special linear group is the kernel of the determinant, example, no evidence levelThe special linear gr…example
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. General linear group object
  2. Group concept
  3. Identity matrix object
  4. Matrix definition
  5. Vector space definition
  6. Group homomorphism definition
  7. Square matrix definition
  8. Linear map definition
  9. Kernel of a homomorphism definition
  10. Multilinear map definition
  11. Alternating multilinear map definition
  12. Determinant definition
  13. Special linear group object
  14. The special linear group is the kernel of the determinant example

Sources

  1. Web https://en.wikipedia.org/wiki/Special_linear_group, lead section

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