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: Bilinear Form
Famous quotes containing the words relation to, relation and/or products:
“You see, I am alive, I am alive
I stand in good relation to the earth
I stand in good relation to the gods
I stand in good relation to all that is beautiful
I stand in good relation to the daughter of Tsen-tainte
You see, I am alive, I am alive”
—N. Scott Momaday (b. 1934)
“The instincts of the ant are very unimportant, considered as the ant’s; but the moment a ray of relation is seen to extend from it to man, and the little drudge is seen to be a monitor, a little body with a mighty heart, then all its habits, even that said to be recently observed, that it never sleeps, become sublime.”
—Ralph Waldo Emerson (1803–1882)
“All that is told of the sea has a fabulous sound to an inhabitant of the land, and all its products have a certain fabulous quality, as if they belonged to another planet, from seaweed to a sailor’s yarn, or a fish story. In this element the animal and vegetable kingdoms meet and are strangely mingled.”
—Henry David Thoreau (1817–1862)