Math 2.0

statement proposition

A linear map is determined by its values on a basis

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.

Explanations

L1 human

A linear map is known as soon as we know where it sends the basis vectors. Every other vector is a combination of basis vectors, and linearity tells us where it goes. Conversely, we may send the basis vectors anywhere we like.

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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 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. Spanning set definition
  8. Basis definition
  9. A linear map is determined by its values on a basis statement

Sources

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

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