Relation To Tensor Products
By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps V ⊗ V → F. If B is a bilinear form on V the corresponding linear map is given by
- v ⊗ w ↦ B(v, w)
The set of all linear maps V ⊗ V → F is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of
- (V ⊗ V)* ≅ V* ⊗ V*
Likewise, symmetric bilinear forms may be thought of as elements of Sym2(V*) (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*).
Read more about this topic: Unimodular Form
Famous quotes containing the words relation to, relation and/or products:
“You must realize that I was suffering from love and I knew him as intimately as I knew my own image in a mirror. In other words, I knew him only in relation to myself.”
—Angela Carter (1940–1992)
“Every word was once a poem. Every new relation is a new word.”
—Ralph Waldo Emerson (1803–1882)
“The reality is that zero defects in products plus zero pollution plus zero risk on the job is equivalent to maximum growth of government plus zero economic growth plus runaway inflation.”
—Dixie Lee Ray (b. 1924)