Math 2.0

statement proposition

Dimension of the annihilator

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 dim⁡W+dim⁡W0=dim⁡V,\dim W + \dim W^0 = \dim V, where W0⊆V∗W^0 \subseteq V^* is the annihilator of WW.

Remarks

A proof extends a basis b1,…,bkb_1, \dots, b_k of WW to a basis b1,…,bnb_1, \dots, b_n of VV. Then the dual basis forms bk+1∗,…,bn∗b_{k+1}^*, \dots, b_n^* are a basis of W0W^0. Another proof uses the isomorphism W0≅(V/W)∗W^0 \cong (V / W)^*. With the double dual isomorphism, it gives (W0)0=W(W^0)^0 = W in finite dimensions.

Explanations

L1 human

The more directions a subspace has, the fewer independent linear forms vanish on it. In R3\R^3, a line is killed by a plane of forms, and a plane is killed by a line of forms. The two dimensions always add up to the dimension of the space.

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To read this node, read these 15 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 map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear form, definition, no evidence levelLinear formdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionAnnihilator of a subspace, definition, no evidence levelAnnihilator of a subs…definitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of the annihilator, statement, no evidence levelDimension of the anni…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 map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear form definition
  7. Linear span definition
  8. Space of linear maps definition
  9. Dual space definition
  10. Spanning set definition
  11. Basis definition
  12. Annihilator of a subspace definition
  13. Finite-dimensional vector space definition
  14. Dimension definition
  15. Dimension of the annihilator statement

Sources

  1. Web https://en.wikipedia.org/wiki/Dual_space, Quotient spaces and annihilators

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