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:
“The foregoing generations beheld God and nature face to face; we, through their eyes. Why should not we also enjoy an original relation to the universe? Why should not we have a poetry and philosophy of insight and not of tradition, and a religion by revelation to us, and not the history of theirs?”
—Ralph Waldo Emerson (1803–1882)
“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)
“But, most of all, the Great Society is not a safe harbor, a resting place, a final objective, a finished work. It is a challenge constantly renewed, beckoning us toward a destiny where the meaning of our lives matches the marvelous products of our labor.”
—Lyndon Baines Johnson (1908–1973)