Math 2.0

Linear and multilinear algebra; matrix theory Definition

Composition of linear maps

Builds on Linear map

Definition

Let FF be a field, let UU, VV, WW be vector spaces over FF, and let S ⁣:U→VS \colon U \to V and T ⁣:V→WT \colon V \to W be linear maps. The composition of TT and SS is the map T∘S ⁣:U→W,(T∘S)(u)=T(S(u)).T \circ S \colon U \to W, \qquad (T \circ S)(u) = T(S(u)).

Remarks

The composition of linear maps is linear. Composition is associative, and the identity maps are its identity elements. The matrix of a composition is the product of the matrices, in the same order. Composition is not commutative: in general T∘S≠S∘TT \circ S \ne S \circ T, even when both are defined.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph
Vector space, definition, no evidence levelVector spacedefinitionLinear map, definition, no evidence levelLinear mapdefinitionComposition of linear maps, definition, no evidence levelComposition of linear…definition
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 map definition
  3. Composition of linear maps definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Linear_map, Definition and first consequences
About this page
Tag05YBKMRVAR
URIm2://05YBKMRVAR
Kinddefinition
WritingW0 stub
MSC15A04
DefinesComposition of linear maps
UsesLinear map

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by