Math 2.0

statement proposition

Dimension of the space of linear maps

Statement

Let FF be a field and let VV and WW be finite-dimensional vector spaces over FF. Then the space of linear maps Hom⁡F(V,W)\Hom_F(V, W) is finite-dimensional, and dim⁡Hom⁡F(V,W)=dim⁡V⋅dim⁡W.\dim \Hom_F(V, W) = \dim V \cdot \dim W.

Remarks

With bases of VV and WW, the map that sends TT to its matrix is an isomorphism Hom⁡F(V,W)≅Mm×n(F)\Hom_F(V, W) \cong M_{m \times n}(F), with n=dim⁡Vn = \dim V and m=dim⁡Wm = \dim W. The matrices EijE_{ij} form a basis of Mm×n(F)M_{m \times n}(F), so its dimension is mnmn. In particular, the dual space of VV has dimension dim⁡V\dim V.

Explanations

L1 human

A linear map from an nn-dimensional space to an mm-dimensional space is an m×nm \times n matrix. A matrix of this size has mnmn free entries. So the space of these maps has dimension mnmn.

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To read this node, read these 12 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 span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionSpanning 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 the space of linear maps, statement, no evidence levelDimension of the spac…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 span definition
  7. Space of linear maps definition
  8. Spanning set definition
  9. Basis definition
  10. Finite-dimensional vector space definition
  11. Dimension definition
  12. Dimension of the space of linear maps statement

Sources

  1. Web https://en.wikipedia.org/wiki/Linear_map, Vector space of linear maps

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