Math 2.0

statement proposition

Dimension of a quotient space

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a subspace of VV. Then the quotient space V/WV / W is finite-dimensional, and dim⁡(V/W)+dim⁡W=dim⁡V,\dim(V / W) + \dim W = \dim V, where dim⁡\dim is the dimension.

Remarks

The number dim⁡(V/W)=dim⁡V−dim⁡W\dim(V / W) = \dim V - \dim W is the codimension of WW in VV. The formula is the rank–nullity theorem for the quotient map V→V/WV \to V / W. For example, a plane WW in R3\R^3 gives dim⁡(R3/W)=1\dim(\R^3 / W) = 1.

Explanations

L1 human

Dividing by a subspace removes its dimensions. What remains has the dimension of the space minus the dimension of the subspace. Ordinary space divided by a plane leaves one dimension.

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To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionQuotient space, definition, no evidence levelQuotient spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of a quotient space, statement, no evidence levelDimension of a quotie…statement
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. Vector space definition
  2. Linear combination definition
  3. Linear subspace definition
  4. Linear independence definition
  5. Linear span definition
  6. Quotient space definition
  7. Spanning set definition
  8. Basis definition
  9. Finite-dimensional vector space definition
  10. Dimension definition
  11. Dimension of a quotient space statement

Sources

  1. Web https://en.wikipedia.org/wiki/Quotient_space_(linear_algebra), Properties

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