Math 2.0

Linear and multilinear algebra; matrix theory Proposition

A linear map is determined by its values on a basis

Builds on BasisLinear map

Statement

Let FF be a field, let VV and WW be vector spaces over FF, and let (bi)i∈I(b_i)_{i \in I} be a basis of VV. If S,T ⁣:V→WS, T \colon V \to W are linear maps with S(bi)=T(bi)S(b_i) = T(b_i) for all i∈Ii \in I, then S=TS = T. Moreover, for each family (wi)i∈I(w_i)_{i \in I} of vectors of WW, there is exactly one linear map T ⁣:V→WT \colon V \to W with T(bi)=wiT(b_i) = w_i for all i∈Ii \in I.

Remarks

The map is T(∑iaibi)=∑iaiwiT(\sum_i a_i b_i) = \sum_i a_i w_i. The values on a basis can be chosen freely; the values on a merely spanning family cannot. This statement is the reason why a linear map between finite-dimensional spaces is described by a matrix.

Reading path

To understand this page from the ground up, read these 9 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 spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionA linear map is determined by its values on a basis, statement, no evidence levelA linear map is deter…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. Spanning set definition
  8. Basis definition
  9. A linear map is determined by its values on a basis statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Basis_(linear_algebra), Properties
About this page
Tag0R8X9XA83A
URIm2://0R8X9XA83A
Kindstatement, proposition
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 0R8X9XA83A.
UsesBasis, Linear map

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