Math 2.0

Linear and multilinear algebra; matrix theory Definition

Image of a linear map

Builds on Linear map

Definition

Let FF be a field, let VV and WW be vector spaces over FF, and let T ⁣:V→WT \colon V \to W be a linear map. The image of TT, also called the range, is im⁡T={T(v):v∈V}⊆W.\im T = \{T(v) : v \in V\} \subseteq W.

Remarks

The image is a linear subspace of WW. For the map x↦Axx \mapsto Ax of a matrix AA, the image is the column space of AA. The map TT is surjective if and only if im⁡T=W\im T = W. The dimension of the image is the rank of TT.

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Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionImage of a linear map, definition, no evidence levelImage of a linear mapdefinition
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  1. Vector space definition
  2. Linear map definition
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Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Image_(mathematics), lead section
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