Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of the space of linear maps

Builds on Space of linear mapsDimension

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.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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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 page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_map, Vector space of linear maps
About this page
Tag0Z9N1ER0W0
URIm2://0Z9N1ER0W0
Kindstatement, proposition
WritingW0 stub
MSC15A03, 15A04
StatusThis copy of the site has no build data. Run m2 status 0Z9N1ER0W0.
UsesSpace of linear maps, Dimension

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