Math 2.0

Linear and multilinear algebra; matrix theory Definition

Product of vector spaces

Builds on Vector space

Definition

Let FF be a field and let VV and WW be vector spaces over FF. The product V×WV \times W is the set of pairs (v,w)(v, w) with v∈Vv \in V and w∈Ww \in W, with the operations (v,w)+(v′,w′)=(v+v′,w+w′),a(v,w)=(av,aw)(v, w) + (v', w') = (v + v', w + w'), \qquad a(v, w) = (av, aw) for v,v′∈Vv, v' \in V, w,w′∈Ww, w' \in W and a∈Fa \in F. It is a vector space over FF with zero vector (0,0)(0, 0).

Remarks

For finitely many spaces, the product is also the external direct sum, written V⊕WV \oplus W. The dimension of a direct product is dim⁡V+dim⁡W\dim V + \dim W. For example, Fm×FnF^m \times F^n is isomorphic to Fm+nF^{m+n}. The projections (v,w)↦v(v, w) \mapsto v and (v,w)↦w(v, w) \mapsto w are linear maps.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Vector space, definition, no evidence levelVector spacedefinitionProduct of vector spaces, definition, no evidence levelProduct of vector spa…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. Product of vector spaces definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_sum_of_modules, Construction for two vector spaces
About this page
Tag0MM085RX00
URIm2://0MM085RX00
Kinddefinition
WritingW1 stated
MSC15A03
DefinesProduct of vector spaces
UsesVector space

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (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